These are second order ode’s solved using Kovacic algorithm. Code is “kovacic”. Number of problems in this table is 3468
|
# |
ODE |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.06 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.185 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.809 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.093 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.616 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.635 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.687 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.519 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.833 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.798 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.676 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.882 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.006 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.575 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.498 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.27 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-15 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.936 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime } = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.956 |
|
|
\[ {}2 y^{\prime \prime }-y^{\prime }-y = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.318 |
|
|
\[ {}4 y^{\prime \prime }+8 y^{\prime }+3 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.314 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.385 |
|
|
\[ {}9 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}6 y^{\prime \prime }-7 y^{\prime }-20 y = 0 \] \(r = {\frac {529}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}35 y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {1681}{4900}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.293 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.148 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }-3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.186 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.728 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.418 |
|
|
\[ {}y^{\prime \prime }+y = 3 x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y^{\prime \prime }-4 y = 12 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.01 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 6 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.842 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 2 x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.994 |
|
|
\[ {}y^{\prime \prime }+2 y = 4 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.594 |
|
|
\[ {}y^{\prime \prime }+2 y = 6 x \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }+2 y = 6 x +4 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.984 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime } = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.945 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-10 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.304 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }+3 y = 0 \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.375 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+5 y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.381 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+25 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.476 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.623 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+4 y = 0 \] \(r = -{\frac {1}{3}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.474 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.906 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }+3 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.474 |
|
|
\[ {}y^{\prime \prime }-i y^{\prime }+6 y = 0 \] \(r = -{\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime } = \left (-2+2 i \sqrt {3}\right ) y \] \(r = -2+2 i \sqrt {3}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.022 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.455 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+25 y = 0 \] \(r = -\frac {65}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.691 |
|
|
\[ {}\frac {x^{\prime \prime }}{2}+3 x^{\prime }+4 x = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.688 |
|
|
\[ {}3 x^{\prime \prime }+30 x^{\prime }+63 x = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.693 |
|
|
\[ {}x^{\prime \prime }+8 x^{\prime }+16 x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.893 |
|
|
\[ {}2 x^{\prime \prime }+12 x^{\prime }+50 x = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.862 |
|
|
\[ {}4 x^{\prime \prime }+20 x^{\prime }+169 x = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.981 |
|
|
\[ {}2 x^{\prime \prime }+16 x^{\prime }+40 x = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{\prime \prime }+10 x^{\prime }+125 x = 0 \] \(r = -100\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.961 |
|
|
\[ {}y^{\prime \prime }+16 y = {\mathrm e}^{3 x} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 3 x +4 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 2 \sin \left (3 x \right ) \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.759 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 3 x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right )^{2} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.785 |
|
|
\[ {}2 y^{\prime \prime }+4 y^{\prime }+7 y = x^{2} \] \(r = -{\frac {5}{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime \prime }-4 y = \sinh \left (x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.197 |
|
|
\[ {}y^{\prime \prime }-4 y = \cosh \left (2 x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.093 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 1+x \,{\mathrm e}^{x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.8 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \cos \left (3 x \right )+3 \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.103 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 x^{2} {\mathrm e}^{3 x}+5 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 x \cos \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.976 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x \left ({\mathrm e}^{-x}-{\mathrm e}^{-2 x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.869 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = x \,{\mathrm e}^{3 x} \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.028 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 x \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.961 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.975 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.267 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.081 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 1+x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.989 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \sin \left (3 x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.503 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (x \right )^{4} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.718 |
|
|
\[ {}y^{\prime \prime }+y = x \cos \left (x \right )^{3} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.723 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 4 \,{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 3 \,{\mathrm e}^{-2 x} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 2 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.689 |
|
|
\[ {}y^{\prime \prime }-4 y = \sinh \left (2 x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.061 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (3 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \sec \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.964 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.841 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.084 |
|
|
\[ {}y^{\prime \prime }-4 y = x \,{\mathrm e}^{x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 72 x^{5} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.015 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{3} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.511 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{4} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.28 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+3 y = 8 x^{\frac {4}{3}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.498 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = \ln \left (x \right ) \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.312 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = x^{2}-1 \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.895 |
|
|
\[ {}x^{\prime \prime }+9 x = 10 \cos \left (2 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{\prime \prime }+4 x = 5 \sin \left (3 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.231 |
|
|
\[ {}x^{\prime \prime }+100 x = 225 \cos \left (5 t \right )+300 \sin \left (5 t \right ) \] \(r = -100\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.939 |
|
|
\[ {}x^{\prime \prime }+25 x = 90 \cos \left (4 t \right ) \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.28 |
|
|
\[ {}m x^{\prime \prime }+k x = F_{0} \cos \left (\omega t \right ) \] \(r = -\frac {k}{m}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.039 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+4 x = 10 \cos \left (3 t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.993 |
|
|
\[ {}x^{\prime \prime }+3 x^{\prime }+5 x = -4 \cos \left (5 t \right ) \] \(r = -{\frac {11}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.577 |
|
|
\[ {}2 x^{\prime \prime }+2 x^{\prime }+x = 3 \sin \left (10 t \right ) \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.009 |
|
|
\[ {}x^{\prime \prime }+3 x^{\prime }+3 x = 8 \cos \left (10 t \right )+6 \sin \left (10 t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.528 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+5 x = 10 \cos \left (3 t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.598 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+13 x = 10 \sin \left (5 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.824 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+13 x = 10 \sin \left (5 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+26 x = 600 \cos \left (10 t \right ) \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.022 |
|
|
\[ {}x^{\prime \prime }+8 x^{\prime }+25 x = 200 \cos \left (t \right )+520 \sin \left (t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.733 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.317 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.315 |
|
|
\[ {}6 y^{\prime \prime }-y^{\prime }-y = 0 \] \(r = {\frac {25}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.344 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }+y = 0 \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.044 |
|
|
\[ {}4 y^{\prime \prime }-9 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.247 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+9 y = 0 \] \(r = {\frac {45}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-2 y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.643 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.707 |
|
|
\[ {}6 y^{\prime \prime }-5 y^{\prime }+y = 0 \] \(r = {\frac {1}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.693 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.002 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+3 y = 0 \] \(r = {\frac {13}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.239 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-4 y = 0 \] \(r = {\frac {33}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.194 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }-9 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.835 |
|
|
\[ {}4 y^{\prime \prime }-y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.715 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.232 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }+y = 0 \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}4 y^{\prime \prime }-y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.939 |
|
|
\[ {}y^{\prime \prime }-\left (2 \alpha -1\right ) y^{\prime }+\alpha \left (\alpha -1\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.376 |
|
|
\[ {}y^{\prime \prime }+\left (3-\alpha \right ) y^{\prime }-2 \left (\alpha -1\right ) y = 0 \] \(r = \frac {1}{4}+\frac {1}{4} \alpha ^{2}+\frac {1}{2} \alpha \) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.503 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }-2 y = 0 \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.424 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+6 y = 0 \] \(r = -5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.645 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-8 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.32 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.542 |
|
|
\[ {}4 y^{\prime \prime }+9 y = 0 \] \(r = -{\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.548 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\frac {5 y}{4} = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.511 |
|
|
\[ {}9 y^{\prime \prime }+9 y^{\prime }-4 y = 0 \] \(r = {\frac {25}{36}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.348 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+\frac {5 y}{4} = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.511 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+\frac {25 y}{4} = 0 \] \(r = -{\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.561 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.127 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.847 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.435 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+\frac {5 y}{4} = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.851 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.181 |
|
|
\[ {}u^{\prime \prime }-u^{\prime }+2 u = 0 \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.443 |
|
|
\[ {}5 u^{\prime \prime }+2 u^{\prime }+7 u = 0 \] \(r = -{\frac {34}{25}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.72 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+6 y = 0 \] \(r = -5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.169 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime }+\left (a^{2}+1\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.953 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.654 |
|
|
\[ {}t^{2} y^{\prime \prime }+4 t y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.901 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+\frac {5 y}{4} = 0 \] \(r = -\frac {1}{2 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.946 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }-6 y = 0 \] \(r = \frac {12}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.383 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.827 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+5 y = 0 \] \(r = -\frac {17}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.928 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.738 |
|
|
\[ {}t^{2} y^{\prime \prime }+7 t y^{\prime }+10 y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.929 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y = 0 \] \(r = \frac {-3 t^{4}+3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.348 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.391 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.425 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }-3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.346 |
|
|
\[ {}4 y^{\prime \prime }+12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.432 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+10 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}4 y^{\prime \prime }+17 y^{\prime }+4 y = 0 \] \(r = {\frac {225}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.353 |
|
|
\[ {}16 y^{\prime \prime }+24 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.435 |
|
|
\[ {}25 y^{\prime \prime }-20 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.431 |
|
|
\[ {}2 y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.524 |
|
|
\[ {}9 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.889 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+82 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.056 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.086 |
|
|
\[ {}4 y^{\prime \prime }+12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.984 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\frac {y}{4} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.734 |
|
|
\[ {}t^{2} y^{\prime \prime }-3 t y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.671 |
|
|
\[ {}t^{2} y^{\prime \prime }+2 t y^{\prime }+\frac {y}{4} = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.64 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-5 t y^{\prime }+5 y = 0 \] \(r = \frac {5}{16 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.15 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.599 |
|
|
\[ {}4 t^{2} y^{\prime \prime }-8 t y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.682 |
|
|
\[ {}t^{2} y^{\prime \prime }+5 t y^{\prime }+13 y = 0 \] \(r = -\frac {37}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.961 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2 \,{\mathrm e}^{t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.577 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 2 \,{\mathrm e}^{-t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 3 \,{\mathrm e}^{-t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 16 \,{\mathrm e}^{\frac {t}{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.956 |
|
|
\[ {}y^{\prime \prime }+9 y = 9 \sec \left (3 t \right )^{2} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.793 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{-2 t}}{t^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.862 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 \csc \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.995 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sec \left (\frac {t}{2}\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.221 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{t}}{t^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.891 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = g \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.067 |
|
|
\[ {}y^{\prime \prime }+4 y = g \left (t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.255 |
|
|
\[ {}t^{2} y^{\prime \prime }-2 y = 3 t^{2}-1 \] \(r = \frac {2}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.947 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 2 t^{3} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.659 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = t^{2} {\mathrm e}^{2 t} \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.428 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 2 \left (-1+t \right )^{2} {\mathrm e}^{-t} \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.265 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \ln \left (x \right ) x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.91 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = g \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.777 |
|
|
\[ {}t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y = 4 t^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.562 |
|
|
\[ {}t^{2} y^{\prime \prime }+7 t y^{\prime }+5 y = t \] \(r = \frac {15}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.5 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = t^{2} {\mathrm e}^{2 t} \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.213 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 2 \left (-1+t \right ) {\mathrm e}^{-t} \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.074 |
|
|
\[ {}u^{\prime \prime }+2 u = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.872 |
|
|
\[ {}u^{\prime \prime }+\frac {u^{\prime }}{4}+2 u = 0 \] \(r = -{\frac {127}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.484 |
|
|
\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (\frac {t}{4}\right ) \] \(r = -{\frac {1023}{256}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.565 |
|
|
\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (2 t \right ) \] \(r = -{\frac {1023}{256}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.287 |
|
|
\[ {}u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u = 3 \cos \left (6 t \right ) \] \(r = -{\frac {1023}{256}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.457 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = f \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.126 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.943 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.709 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.006 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.412 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.379 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.911 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.308 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.304 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.295 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.375 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.236 |
|
|
\[ {}y^{\prime \prime }+9 y = \tan \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.817 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \sec \left (2 x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.581 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \frac {4}{1+{\mathrm e}^{-x}} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.84 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 3 \,{\mathrm e}^{x} \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 14 x^{\frac {3}{2}} {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }-y = \frac {4 \,{\mathrm e}^{-x}}{1-{\mathrm e}^{-2 x}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.871 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 2 x^{2}+2 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.238 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.501 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 4 \,{\mathrm e}^{-x \left (2+x \right )} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.828 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{\frac {5}{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.252 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 2 x^{4} \sin \left (x \right ) \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.786 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = \left (2 x +1\right )^{2} {\mathrm e}^{-x} \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.445 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.0 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y = x^{1+a} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
21.075 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = x^{3} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.002 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-4 x^{3} y = 8 x^{5} \] \(r = \frac {16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.533 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 8 x^{\frac {5}{2}} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.745 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = x^{\frac {7}{2}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.87 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 3 x^{4} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.772 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = x^{3} {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.306 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = x^{\frac {3}{2}} \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.704 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +4\right ) y^{\prime }+2 \left (x +3\right ) y = x^{4} {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.997 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 2 x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.537 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = x^{4} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.7 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 2 \left (-1+x \right )^{2} {\mathrm e}^{x} \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.357 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = x^{\frac {5}{2}} {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.039 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y = \left (3 x -1\right )^{2} {\mathrm e}^{2 x} \] \(r = \frac {81 x^{2}-108 x +54}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.125 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }+2 y = \left (-1+x \right )^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.535 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 2 x \] \(r = \frac {4 x}{\left (-1+x \right )^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 3\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
16.8 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = -2 x^{2} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.133 |
|
|
\[ {}\left (1+x \right ) \left (2 x +3\right ) y^{\prime \prime }+2 \left (2+x \right ) y^{\prime }-2 y = \left (2 x +3\right )^{2} \] \(r = \frac {3}{\left (2 x +3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.083 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] \(r = \frac {5}{16 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.391 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] \(r = \frac {5}{16 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.299 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y = 0 \] \(r = \frac {t^{2}}{4}-\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.205 |
|
|
\[ {}y^{\prime \prime }+t y^{\prime }+y = 0 \] \(r = \frac {t^{2}}{4}-\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.416 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.351 |
|
|
\[ {}6 y^{\prime \prime }-7 y^{\prime }+y = 0 \] \(r = {\frac {25}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.503 |
|
|
\[ {}3 y^{\prime \prime }+6 y^{\prime }+3 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-4 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.84 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-10 y = 0 \] \(r = {\frac {81}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.158 |
|
|
\[ {}5 y^{\prime \prime }+5 y^{\prime }-y = 0 \] \(r = {\frac {9}{20}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.638 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+y = 0 \] \(r = 8\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.933 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.744 |
|
|
\[ {}t^{2} y^{\prime \prime }+\alpha t y^{\prime }+\beta y = 0 \] \(r = \frac {\alpha ^{2}-2 \alpha -4 \beta }{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.751 |
|
|
\[ {}t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y = 0 \] \(r = \frac {35}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.38 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }-2 y = 0 \] \(r = \frac {11}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
6.679 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+4 y = 0 \] \(r = -3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.17 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.845 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }+4 y = 0 \] \(r = -{\frac {23}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+3 y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.82 |
|
|
\[ {}4 y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = -{\frac {15}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.888 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+2 y = 0 \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.162 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.177 |
|
|
\[ {}2 y^{\prime \prime }-y^{\prime }+3 y = 0 \] \(r = -{\frac {23}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.005 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }+4 y = 0 \] \(r = -{\frac {11}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.797 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.204 |
|
|
\[ {}t^{2} y^{\prime \prime }+2 t y^{\prime }+2 y = 0 \] \(r = -\frac {2}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.766 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.486 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.532 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.116 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.093 |
|
|
\[ {}9 y^{\prime \prime }-12 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] \(r = \frac {6}{\left (t^{2}+2 t -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.399 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.315 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = \frac {2 t^{2}-3}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
2.775 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.386 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] \(r = \frac {6 t^{2}-7}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
1.505 |
|
|
\[ {}\left (1+2 t \right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y = 0 \] \(r = \frac {4 t^{2}+2}{\left (1+2 t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.343 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.497 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.682 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.765 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.048 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = t \,{\mathrm e}^{2 t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.849 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }+y = \left (t^{2}+1\right ) {\mathrm e}^{t} \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.963 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = t \,{\mathrm e}^{3 t}+1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.8 |
|
|
\[ {}3 y^{\prime \prime }+4 y^{\prime }+y = \sin \left (t \right ) {\mathrm e}^{-t} \] \(r = {\frac {1}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.539 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = t^{\frac {5}{2}} {\mathrm e}^{-2 t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.86 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \sqrt {t +1} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.749 |
|
|
\[ {}y^{\prime \prime }-y = f \left (t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.052 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = t^{2}+1 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.929 |
|
|
\[ {}m y^{\prime \prime }+c y^{\prime }+k y = F_{0} \cos \left (\omega t \right ) \] \(r = \frac {c^{2}-4 k m}{4 m^{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.87 |
|
|
\[ {}t^{2} y^{\prime \prime }-5 t y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.365 |
|
|
\[ {}t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y = 0 \] \(r = \frac {35}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.343 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+3 t y^{\prime }-y = 0 \] \(r = \frac {5}{16 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.304 |
|
|
\[ {}\left (-1+t \right )^{2} y^{\prime \prime }-2 \left (-1+t \right ) y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.893 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.58 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.729 |
|
|
\[ {}\left (t -2\right )^{2} y^{\prime \prime }+5 \left (t -2\right ) y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 \left (t -2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.547 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.097 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+2 y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
4.398 |
|
|
\[ {}t^{2} y^{\prime \prime }-3 t y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.281 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.031 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+12 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.438 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.471 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }-2 y = 0 \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.454 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-y = 0 \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-2 y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.506 |
|
|
\[ {}2 y^{\prime \prime }+2 y^{\prime }-y = 0 \] \(r = {\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.521 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.107 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+3 y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime }-4 y = 3 \cos \left (x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.035 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.531 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = {\mathrm e}^{x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{-2 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.722 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.404 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.791 |
|
|
\[ {}y^{\prime \prime }-4 y = x +{\mathrm e}^{2 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.954 |
|
|
\[ {}y^{\prime \prime }-9 y = {\mathrm e}^{3 x}+\sin \left (3 x \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.813 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = x^{3} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.785 |
|
|
\[ {}-2 y^{\prime \prime }+3 y = x \,{\mathrm e}^{x} \] \(r = {\frac {3}{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.006 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.862 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{x} \sin \left (3 x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.296 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = x^{3} {\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime \prime }+2 n y^{\prime }+n^{2} y = 5 \cos \left (6 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.63 |
|
|
\[ {}y^{\prime \prime }+9 y = \left (1+\sin \left (3 x \right )\right ) \cos \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.905 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 2 x -{\mathrm e}^{-4 x}+\sin \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.204 |
|
|
\[ {}y^{\prime \prime }+4 y = 8 \sin \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.803 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }-6 y = {\mathrm e}^{3 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime \prime }+4 y = 12 \cos \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.716 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.248 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.968 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime } = 8 \sin \left (2 x \right )+{\mathrm e}^{-x} \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
8.673 |
|
|
\[ {}y^{\prime \prime }+y = 3 x \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.874 |
|
|
\[ {}2 y^{\prime \prime }+5 y^{\prime }-3 y = \sin \left (x \right )-8 x \] \(r = {\frac {49}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.78 |
|
|
\[ {}8 y^{\prime \prime }-y = x \,{\mathrm e}^{-\frac {x}{2}} \] \(r = {\frac {1}{8}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.313 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.157 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.832 |
|
|
\[ {}y^{\prime \prime }+4 y = x^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.096 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.801 |
|
|
\[ {}y^{\prime \prime }+y = 4 \sin \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.109 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 x -2 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.832 |
|
|
\[ {}y^{\prime \prime }-y = 3 x +5 \,{\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.976 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{x}+\sin \left (4 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.283 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.227 |
|
|
\[ {}y^{\prime \prime }+a^{2} y = \sec \left (x a \right ) \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.131 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{\left (1-x \right )^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.966 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \sin \left ({\mathrm e}^{-x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.23 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (x \right ) \tan \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.887 |
|
|
\[ {}y^{\prime \prime }-2 y = {\mathrm e}^{-x} \sin \left (2 x \right ) \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.701 |
|
|
\[ {}y^{\prime \prime }+9 y = \sec \left (x \right ) \csc \left (x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.156 |
|
|
\[ {}y^{\prime \prime }+9 y = \csc \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.699 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (\frac {x}{3}\right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.701 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = {\mathrm e}^{\frac {x}{2}} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 \,{\mathrm e}^{x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}y^{\prime \prime }+3 y = 3 \,{\mathrm e}^{-4 x} \] \(r = -3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.374 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.056 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = {\mathrm e}^{-2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}y^{\prime \prime }+2 y = \sin \left (x \right ) \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.934 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{3 x}}{2}-\frac {{\mathrm e}^{-3 x}}{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-2 y = \sin \left (2 x \right ) \] \(r = {\frac {17}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.373 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.971 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{3 x} \left (1+\sin \left (2 x \right )\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.901 |
|
|
\[ {}y^{\prime \prime }+2 n^{2} y^{\prime }+n^{4} y = \sin \left (k x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.6 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.228 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = x \,{\mathrm e}^{-x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+4 y = x \,{\mathrm e}^{x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.066 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+2 y = {\mathrm e}^{-x} x^{2} \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = x^{2}-8 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.002 |
|
|
\[ {}y^{\prime \prime }+y = x^{2} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.452 |
|
|
\[ {}y^{\prime \prime }-y = x^{2} \cos \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.287 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }-2 y = x^{2} {\mathrm e}^{x} \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.866 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x^{2} \cos \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.406 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = x^{2} \sin \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.451 |
|
|
\[ {}y^{\prime \prime }-y = x \sin \left (2 x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.333 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = x^{3} \sin \left (2 x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
19.685 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = x \,{\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
7.667 |
|
|
\[ {}y^{\prime \prime }-4 y = x \,{\mathrm e}^{2 x} \cos \left (x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.38 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = x^{2} {\mathrm e}^{-x} \sin \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
14.476 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+y = 0 \] \(r = \frac {5}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
6.27 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+16 y = 0 \] \(r = -\frac {65}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.542 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-16 x y^{\prime }+25 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.653 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+10 y = 0 \] \(r = -\frac {25}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.26 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x y^{\prime }-18 y = \ln \left (x \right ) \] \(r = \frac {165}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.647 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x y^{\prime }+2 y = \ln \left (x^{2}\right ) \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
9.288 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{3} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.391 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 1-x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.723 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 4 x +\sin \left (\ln \left (x \right )\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
30.744 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = \ln \left (x \right ) x^{2} \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.062 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+3 y = \left (-1+x \right ) \ln \left (x \right ) \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
308.145 |
|
|
\[ {}y^{\prime \prime } = \cos \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.969 |
|
|
\[ {}y^{\prime \prime } = k^{2} y \] \(r = k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.309 |
|
|
\[ {}x^{\prime \prime }+k^{2} x = 0 \] \(r = -k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.514 |
|
|
\[ {}x y^{\prime \prime } = x^{2}+1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.438 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime } = y^{\prime } \] \(r = -\frac {1}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.671 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (y^{\prime }+1\right ) = 0 \] \(r = \frac {1}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.365 |
|
|
\[ {}x y^{\prime \prime }+x = y^{\prime } \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.142 |
|
|
\[ {}x^{\prime \prime }+t x^{\prime } = t^{3} \] \(r = \frac {t^{2}}{4}+\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.288 |
|
|
\[ {}x^{2} y^{\prime \prime } = x y^{\prime }+1 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.694 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime } = 1 \] \(r = \frac {3 x^{2}+2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.097 |
|
|
\[ {}y^{\prime \prime } = y \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.687 |
|
|
\[ {}y^{\prime \prime } = \sec \left (x \right ) \tan \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
3.814 |
|
|
\[ {}x^{\prime \prime }-k^{2} x = 0 \] \(r = k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.021 |
|
|
\[ {}x^{\prime \prime }+\omega _{0}^{2} x = a \cos \left (\omega t \right ) \] \(r = -\omega _{0}^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.99 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}f^{\prime \prime }+2 f^{\prime }+5 f = {\mathrm e}^{-t} \cos \left (3 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}f^{\prime \prime }+6 f^{\prime }+9 f = {\mathrm e}^{-t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.654 |
|
|
\[ {}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.59 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.407 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.502 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+3 \left (1+x \right ) y^{\prime }+y = x^{2} \] \(r = -\frac {1}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.731 |
|
|
\[ {}\left (-2+x \right ) y^{\prime \prime }+3 y^{\prime }+\frac {4 y}{x^{2}} = 0 \] \(r = \frac {3 x^{2}-16 x +32}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-y = x^{n} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.429 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = {\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.322 |
|
|
\[ {}y^{\prime \prime }-25 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.183 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.619 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.514 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.855 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.283 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 9 x^{2} \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.592 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4} \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.773 |
|
|
\[ {}y^{\prime \prime }-\left (a +b \right ) y^{\prime }+y a b = 0 \] \(r = \frac {1}{4} a^{2}-\frac {1}{2} a b +\frac {1}{4} b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.224 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+\left (a^{2}+b^{2}\right ) y = 0 \] \(r = -b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.247 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.178 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.242 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.474 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime } = x^{n} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime } = \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.194 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-8 y = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.826 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \ln \left (x \right ) x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.636 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 9 x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.858 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.189 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.2 |
|
|
\[ {}y^{\prime \prime }-36 y = 0 \] \(r = 36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.61 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }-8 y = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.77 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.52 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 18 \,{\mathrm e}^{5 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 4 x^{2}+5 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{\prime \prime }+y = 6 \,{\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 5 \,{\mathrm e}^{-2 x} x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.452 |
|
|
\[ {}y^{\prime \prime }+4 y = 8 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.515 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \,{\mathrm e}^{2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.413 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 3 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+9 y = 5 \cos \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.797 |
|
|
\[ {}y^{\prime \prime }-y = 9 \,{\mathrm e}^{2 x} x \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = -10 \sin \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 4 \cos \left (x \right )-2 \sin \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }+\omega ^{2} y = \frac {F_{0} \cos \left (\omega t \right )}{m} \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.726 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+6 y = 7 \,{\mathrm e}^{2 x} \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.643 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = \sin \left (x \right )^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+6 y = \sin \left (x \right )^{2} \cos \left (x \right )^{2} \] \(r = -6\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.912 |
|
|
\[ {}y^{\prime \prime }-16 y = 20 \cos \left (4 x \right ) \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.481 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 50 \sin \left (3 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }-y = 10 \,{\mathrm e}^{2 x} \cos \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 169 \sin \left (3 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.605 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 40 \sin \left (x \right )^{2} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime \prime }+y = 3 \,{\mathrm e}^{x} \cos \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 2 \,{\mathrm e}^{-x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime }-4 y = 100 x \,{\mathrm e}^{x} \sin \left (x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \,{\mathrm e}^{-x} \cos \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+10 y = 24 \,{\mathrm e}^{x} \cos \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }+16 y = 34 \,{\mathrm e}^{x}+16 \cos \left (4 x \right )-8 \sin \left (4 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.276 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 4 \,{\mathrm e}^{3 x} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.565 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{-2 x}}{x^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }+9 y = 18 \sec \left (3 x \right )^{3} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = \frac {2 \,{\mathrm e}^{-3 x}}{x^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.597 |
|
|
\[ {}y^{\prime \prime }-4 y = \frac {8}{{\mathrm e}^{2 x}+1} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.544 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+9 y = \frac {36}{4-\cos \left (3 x \right )^{2}} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = \frac {2 \,{\mathrm e}^{5 x}}{x^{2}+4} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 4 \,{\mathrm e}^{3 x} \sec \left (2 x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.822 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )+4 \,{\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.698 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right )+2 x^{2}+5 x +1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.716 |
|
|
\[ {}y^{\prime \prime }-y = 2 \tanh \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime \prime }-2 m y^{\prime }+m^{2} y = \frac {{\mathrm e}^{m x}}{x^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.572 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {4 \,{\mathrm e}^{x} \ln \left (x \right )}{x^{3}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \frac {{\mathrm e}^{-x}}{\sqrt {-x^{2}+4}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.606 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+17 y = \frac {64 \,{\mathrm e}^{-x}}{3+\sin \left (4 x \right )^{2}} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.911 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {4 \,{\mathrm e}^{-2 x}}{x^{2}+1}+2 x^{2}-1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 15 \,{\mathrm e}^{-2 x} \ln \left (x \right )+25 \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.893 |
|
|
\[ {}y^{\prime \prime }-9 y = F \left (x \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = F \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.639 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = F \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-12 y = F \left (x \right ) \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 5 \,{\mathrm e}^{2 x} x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.798 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 4 \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.421 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.505 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 9 \ln \left (x \right ) \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.307 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+5 y = 8 x \ln \left (x \right )^{2} \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.878 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4} \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.737 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+6 y = 4 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.735 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = \frac {x^{2}}{\ln \left (x \right )} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.603 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 m -1\right ) x y^{\prime }+m^{2} y = x^{m} \ln \left (x \right )^{k} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
9.447 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+5 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.285 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+25 y = 0 \] \(r = -\frac {101}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 4 \,{\mathrm e}^{-3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 4 \,{\mathrm e}^{-2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime \prime }-4 y = 5 \,{\mathrm e}^{x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.35 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 x \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }-y = 4 \,{\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.389 |
|
|
\[ {}y^{\prime \prime }+4 y = \ln \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.698 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 5 \,{\mathrm e}^{x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.412 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+y = 4 \cos \left (2 x \right )+3 \,{\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+10 y = 3 x \,{\mathrm e}^{-3 x}-2 \,{\mathrm e}^{3 x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.997 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+17 y = {\mathrm e}^{4 x} \left (x^{2}-3 x \sin \left (x \right )\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.506 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = \left (x +{\mathrm e}^{x}\right ) \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.297 |
|
|
\[ {}y^{\prime \prime }+4 y = \sinh \left (x \right ) \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.947 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \cosh \left (x \right ) \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.684 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.281 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.059 |
|
|
\[ {}6 y^{\prime \prime }-11 y^{\prime }+4 y = 0 \] \(r = {\frac {25}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-y = 0 \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }-2 y = 0 \] \(r = k^{2}+2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.387 |
|
|
\[ {}y^{\prime \prime }+4 k y^{\prime }-12 k^{2} y = 0 \] \(r = 16 k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.437 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.344 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.34 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.815 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.702 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.736 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.279 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 4 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.433 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{i x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.761 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.567 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 8+6 \,{\mathrm e}^{x}+2 \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.218 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 9 x \,{\mathrm e}^{x}+10 \,{\mathrm e}^{-x} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 2 \,{\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.536 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}+2 x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.964 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x +\sin \left (2 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.503 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} x^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.862 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{-2 x}+x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = x +{\mathrm e}^{2 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.683 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )+{\mathrm e}^{-x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.292 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.139 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (2 x \right ) \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.018 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }-6 y = {\mathrm e}^{3 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \sin \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.852 |
|
|
\[ {}y^{\prime \prime }+9 y = 8 \cos \left (x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = {\mathrm e}^{x} \left (2 x -3\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right )^{2} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.808 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} x^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.574 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.709 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.672 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.97 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \frac {{\mathrm e}^{-x}}{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{x} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.708 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \cos \left ({\mathrm e}^{-x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.796 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.695 |
|
|
\[ {}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\frac {2 y}{x^{2}} = x \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.991 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x^{3} \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.967 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = {\mathrm e}^{-x} x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.137 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = \frac {1}{x} \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.844 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.115 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.628 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (y^{\prime }+1\right ) = 0 \] \(r = \frac {1}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.169 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.986 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.595 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.842 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.526 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] \(r = \frac {-a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {a^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.542 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {-n^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.81 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] \(r = \frac {2 a^{2}-x^{2}}{x^{2} a^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.871 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] \(r = \frac {q^{2} x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.484 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.524 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.303 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.319 |
|
|
\[ {}y^{\prime \prime }+9 y^{\prime } = 0 \] \(r = {\frac {81}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.316 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+6 y = 0 \] \(r = -5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.353 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.668 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.713 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.332 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-y = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+\left (1+2 i\right ) y^{\prime }+\left (-1+i\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.311 |
|
|
\[ {}y^{\prime \prime }+\left (1+2 i\right ) y^{\prime }+\left (-1+i\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.29 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 10 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 16 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = {\mathrm e}^{2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 24 \,{\mathrm e}^{-3 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.41 |
|
|
\[ {}y^{\prime \prime }+y = 2 \,{\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.409 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 12 \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 3 \,{\mathrm e}^{2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }-16 y = 40 \,{\mathrm e}^{4 x} \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 6 \,{\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.595 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 100 \cos \left (4 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+12 y = 80 \sin \left (2 x \right ) \] \(r = -8\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.925 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+25 y = 120 \sin \left (5 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.773 |
|
|
\[ {}5 y^{\prime \prime }+12 y^{\prime }+20 y = 120 \sin \left (2 x \right ) \] \(r = -{\frac {64}{25}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.889 |
|
|
\[ {}y^{\prime \prime }+9 y = 30 \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.637 |
|
|
\[ {}y^{\prime \prime }+16 y = 16 \cos \left (4 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.608 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+17 y = 60 \,{\mathrm e}^{-4 x} \sin \left (5 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.769 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+5 y = 40 \,{\mathrm e}^{-\frac {3 x}{2}} \sin \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+8 y = 30 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {5 x}{2}\right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.858 |
|
|
\[ {}5 y^{\prime \prime }+6 y^{\prime }+2 y = x^{2}+6 x \] \(r = -{\frac {1}{25}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime } = 2 x \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.31 |
|
|
\[ {}y^{\prime \prime }+y = 2 x \,{\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 12 x \,{\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 16 \,{\mathrm e}^{-x} x^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.565 |
|
|
\[ {}y^{\prime \prime }+y = 8 x \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }+y = x^{3}-1+2 \cos \left (x \right )+\left (2-4 x \right ) {\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.52 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2 \,{\mathrm e}^{x}+6 x -5 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.557 |
|
|
\[ {}y^{\prime \prime }-y = \sinh \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.751 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (x \right )+4 x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{x}+\left (1-x \right ) \left ({\mathrm e}^{2 x}-1\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.767 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 9 x \,{\mathrm e}^{-x}-6 x^{2}+4 \,{\mathrm e}^{2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.269 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime } = 0 \] \(r = x^{2}+1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.459 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.545 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.941 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.026 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+6 y = 0 \] \(r = -\frac {21}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.272 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-16 y = 8 x^{4} \] \(r = \frac {63}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.56 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x -\frac {1}{x} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.462 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 2 x^{3} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.783 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 6 \ln \left (x \right ) x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+y = 3 x^{2} \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.57 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 2 x \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.932 |
|
|
\[ {}r^{\prime \prime }-6 r^{\prime }+9 r = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.317 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 10 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{-x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.824 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.896 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 4 x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.035 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 26 \,{\mathrm e}^{3 x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.477 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 2 \,{\mathrm e}^{-2 x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 6 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = {\mathrm e}^{2 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 5 x +4 \,{\mathrm e}^{x} \left (1+\sin \left (2 x \right )\right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.926 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 6 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime } = -4 y \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.68 |
|
|
\[ {}y^{\prime \prime } = y \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.131 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.998 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}x^{\prime \prime }-\omega ^{2} x = 0 \] \(r = \omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.794 |
|
|
\[ {}x^{\prime \prime }+42 x^{\prime }+x = 0 \] \(r = 440\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.707 |
|
|
\[ {}x^{\prime \prime }+2 \gamma x^{\prime }+\omega _{0} x = F \cos \left (\omega t \right ) \] \(r = \gamma ^{2}-\omega _{0}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.78 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.086 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.141 |
|
|
\[ {}y^{\prime \prime }+16 y = 16 \cos \left (4 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime \prime }-y = \cosh \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.301 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.64 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+2 \left (1-2 x \right ) y^{\prime }-2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.774 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.366 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+2 y = 0 \] \(r = \frac {-3-32 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.381 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.588 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.718 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {x +8}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.466 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{\left (-1+x \right )^{2} x}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 8 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.359 |
|
|
\[ {}y^{\prime \prime }-4 y = 10 \,{\mathrm e}^{3 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.385 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.441 |
|
|
\[ {}y^{\prime \prime }+25 y = 5 x^{2}+x \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 4 \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 2 \,{\mathrm e}^{-2 x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.164 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }-y = 2 x -3 \] \(r = {\frac {4}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.41 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = 8 \,{\mathrm e}^{4 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.446 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }-4 y = {\mathrm e}^{3 x} \] \(r = {\frac {81}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.399 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 54 x +18 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 100 \sin \left (4 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.511 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 4 \sinh \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.141 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 2 \cosh \left (2 x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.29 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+10 y = 20-{\mathrm e}^{2 x} \] \(r = -{\frac {39}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \cos \left (x \right )^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.655 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = x +{\mathrm e}^{2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+3 y = x^{2}-1 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }-9 y = {\mathrm e}^{3 x}+\sin \left (x \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.185 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+3 x = {\mathrm e}^{-3 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 6 \sin \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.54 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }+2 x = \sin \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.631 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 3 \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.637 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+10 y = 50 x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.528 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+2 x = 85 \sin \left (3 t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.38 |
|
|
\[ {}y^{\prime \prime } = 3 \sin \left (x \right )-4 y \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.727 |
|
|
\[ {}\frac {x^{\prime \prime }}{2} = -48 x \] \(r = -96\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.251 |
|
|
\[ {}x^{\prime \prime }+5 x^{\prime }+6 x = \cos \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 4 x^{2} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.406 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.355 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = \sin \left (2 x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 2 \sin \left (\frac {t}{2}\right )-\cos \left (\frac {t}{2}\right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.12 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 64 \,{\mathrm e}^{-t} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 50 t^{3}-36 t^{2}-63 t +18 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime } = 9 x^{2}+2 x -1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.609 |
|
|
\[ {}y^{\prime \prime }-5 y = 2 \,{\mathrm e}^{5 x} \] \(r = 5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x^{2}-1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 4 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 4 \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.521 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 3 \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.43 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.441 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.964 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.356 |
|
|
\[ {}x^{\prime \prime }+4 x = \sin \left (2 t \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.809 |
|
|
\[ {}t^{2} N^{\prime \prime }-2 t N^{\prime }+2 N = \ln \left (t \right ) t \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.818 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{x^{5}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.959 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.994 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}y^{\prime \prime }-60 y^{\prime }-900 y = 5 \,{\mathrm e}^{10 x} \] \(r = 1800\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime } = -3 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.546 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = \ln \left (x \right ) \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.651 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime } = x^{3} {\mathrm e}^{x} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.254 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.271 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.739 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.613 |
|
|
\[ {}y^{\prime \prime }-y = 4-x \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.348 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.238 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 2 \left (1-x \right ) {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.237 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{5 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }+9 y = x \cos \left (x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.829 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.108 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.246 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-15 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.266 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.273 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.436 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 1 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.358 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 5 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.078 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = -2 x^{2}+2 x +2 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.382 |
|
|
\[ {}y^{\prime \prime }-y = 4 x \,{\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.445 |
|
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right )^{2} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.66 |
|
|
\[ {}y^{\prime \prime }-y = \frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.524 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \sin \left ({\mathrm e}^{-x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.575 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+4 y = 4 \sec \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.807 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = \frac {1}{1+{\mathrm e}^{-x}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{-x} \sin \left ({\mathrm e}^{-x}\right )+\cos \left ({\mathrm e}^{-x}\right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.073 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }-y = \frac {1}{\left (1+{\mathrm e}^{-x}\right )^{2}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.463 |
|
|
\[ {}y^{\prime \prime }+2 y = 2+{\mathrm e}^{x} \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.614 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{x} \sin \left (2 x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.612 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = x^{2}+\sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }-9 y = x +{\mathrm e}^{2 x}-\sin \left (2 x \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime }+y = -2 \sin \left (x \right )+4 x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-2 x}+5 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.391 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{x}+{\mathrm e}^{2 x} x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.552 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime }+5 y = \cos \left (\sqrt {5}\, x \right ) \] \(r = -5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.73 |
|
|
\[ {}y^{\prime \prime }-y = x^{2} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime \prime }+2 y = x^{3}+x^{2}+{\mathrm e}^{-2 x}+\cos \left (3 x \right ) \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.331 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-y = {\mathrm e}^{x} \cos \left (x \right ) \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \frac {{\mathrm e}^{2 x}}{x^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.534 |
|
|
\[ {}y^{\prime \prime }-y = x \,{\mathrm e}^{3 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.404 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{-2 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.136 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x +\ln \left (x \right ) x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.908 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = \ln \left (x \right )^{2}-\ln \left (x^{2}\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
11.81 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = \ln \left (1+x \right )^{2}+x -1 \] \(r = \frac {3}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.375 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y = 6 x \] \(r = \frac {15}{\left (2 x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
1.691 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 2 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.546 |
|
|
\[ {}\left (x^{2}+4\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 8 \] \(r = -\frac {12}{\left (x^{2}+4\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.884 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+\left (2+x \right ) y = \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x} \] \(r = \frac {3}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+\left (x^{2}+3 x +3\right ) y = \left (-x^{2}+6\right ) {\mathrm e}^{x} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.617 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right )^{2} y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.719 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = \left (x^{2}-x +1\right ) {\mathrm e}^{x} \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.564 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.25 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+y = \frac {1+x}{x} \] \(r = -\frac {1}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.923 |
|
|
\[ {}x^{8} y^{\prime \prime }+4 x^{7} y^{\prime }+y = \frac {1}{x^{3}} \] \(r = \frac {2 x^{6}-1}{x^{8}}\) \(L = [1]\) case used \(1\) poles order = \([8]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.064 |
|
|
\[ {}x y^{\prime \prime }-3 y^{\prime }+\frac {3 y}{x} = 2+x \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.818 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (3 x +4\right ) y^{\prime }+3 y = \left (3 x +2\right ) {\mathrm e}^{3 x} \] \(r = \frac {9 x^{2}+12 x +6}{4 \left (1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.077 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (9 x^{2}+6\right ) y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 4 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.108 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \frac {-x^{2}+1}{x} \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.576 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {2}{x^{3}} \] \(r = \frac {1}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.133 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = -\frac {2}{x}-\ln \left (x \right ) \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-y = 0 \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.218 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = \frac {-5 x^{2}-2}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.136 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] \(r = -\frac {2}{x^{2} \left (x^{2}-1\right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.171 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {3 x^{2}-6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.717 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] \(r = -\frac {4}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.605 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = \frac {-5 x^{2}-2}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 2 x \,{\mathrm e}^{x}-1 \] \(r = \frac {x^{2}}{4}-\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.059 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x \] \(r = \frac {x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.22 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{2}+2 x \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.867 |
|
|
\[ {}x^{3} y^{\prime \prime }+x y^{\prime }-y = \cos \left (\frac {1}{x}\right ) \] \(r = \frac {1}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.593 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }-y = x +\frac {1}{x} \] \(r = \frac {3}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.536 |
|
|
\[ {}2 x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-y = x^{2}-1 \] \(r = \frac {x^{2}+4 x +12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.359 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\frac {y}{4} = -\frac {x^{2}}{2}+\frac {1}{2} \] \(r = -\frac {3}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
6.195 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.223 |
|
|
\[ {}s^{\prime \prime }+2 s^{\prime }+s = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.617 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.486 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 1+3 x \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.378 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{2 x} x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.393 |
|
|
\[ {}y^{\prime \prime }+y = 4 \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u = f \left (x \right ) \] \(r = \frac {-2 p q -4 p r +q^{2}}{4 x^{2} p^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.53 |
|
|
\[ {}\sin \left (x \right ) u^{\prime \prime }+2 \cos \left (x \right ) u^{\prime }+\sin \left (x \right ) u = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
1.17 |
|
|
\[ {}y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1} = 0 \] \(r = \frac {3 x^{2}-6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.795 |
|
|
\[ {}u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 50 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 50 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (2 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+4 y = x^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.6 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = x^{3} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.389 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] \(r = -\frac {2}{\left (1+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.995 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] \(r = \frac {2}{\left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.504 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.61 |
|
|
\[ {}y^{\prime \prime } = 2+x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.848 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.159 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = 0 \] \(r = -k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.375 |
|
|
\[ {}y^{\prime \prime } = 1+3 x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.197 |
|
|
\[ {}3 y^{\prime \prime }+2 y = 0 \] \(r = -{\frac {2}{3}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.757 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.253 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+2 i y^{\prime }+y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y = 0 \] \(r = -2+\frac {3 i}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.33 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.95 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.652 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.816 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.473 |
|
|
\[ {}y^{\prime \prime }+\left (1+4 i\right ) y^{\prime }+y = 0 \] \(r = -\frac {19}{4}+2 i\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.129 |
|
|
\[ {}y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y = 0 \] \(r = -2+\frac {3 i}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.533 |
|
|
\[ {}y^{\prime \prime }+10 y = 0 \] \(r = -10\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.344 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.658 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.788 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+2 i y^{\prime }+y = x \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 3 \,{\mathrm e}^{-x}+2 x^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.666 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = \sin \left (x \right ) \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.503 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (2 x \right ) \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.153 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.475 |
|
|
\[ {}4 y^{\prime \prime }-y = {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}6 y^{\prime \prime }+5 y^{\prime }-6 y = x \] \(r = {\frac {169}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.44 |
|
|
\[ {}y^{\prime \prime }+\omega ^{2} y = A \cos \left (\omega x \right ) \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.286 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.71 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.634 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }-y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.365 |
|
|
\[ {}y^{\prime \prime }-2 i y^{\prime }-y = {\mathrm e}^{i x}-2 \,{\mathrm e}^{-i x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.722 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.48 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }-4 y = 3 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{-x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.568 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = x^{2}+\cos \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime }+9 y = x^{2} {\mathrm e}^{3 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.714 |
|
|
\[ {}y^{\prime \prime }+y = x \,{\mathrm e}^{x} \cos \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.089 |
|
|
\[ {}y^{\prime \prime }+i y^{\prime }+2 y = 2 \cosh \left (2 x \right )+{\mathrm e}^{-2 x} \] \(r = -{\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.957 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.038 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.88 |
|
|
\[ {}\left (3 x -1\right )^{2} y^{\prime \prime }+\left (9 x -3\right ) y^{\prime }-9 y = 0 \] \(r = \frac {27}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.356 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.83 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\alpha ^{2} y = 0 \] \(r = \frac {4 \alpha ^{2} x^{2}-4 \alpha ^{2}-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.593 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.141 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.513 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.326 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.587 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 1 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.092 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+5 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.629 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (-2-i\right ) x y^{\prime }+3 i y = 0 \] \(r = \frac {\frac {7}{4}-\frac {3 i}{2}}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.288 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 \pi y = x \] \(r = \frac {16 \pi -1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.379 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.278 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = 0 \] \(r = -k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.828 |
|
|
\[ {}x y^{\prime \prime }-2 y^{\prime } = x^{3} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.16 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.396 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.219 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+4 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.26 |
|
|
\[ {}y^{\prime \prime }-k^{2} y = 0 \] \(r = k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.232 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 4 x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.368 |
|
|
\[ {}x y^{\prime \prime }-3 y^{\prime } = 5 x \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
6.202 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.399 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+8 y = 0 \] \(r = -8\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.45 |
|
|
\[ {}2 y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.735 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+20 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.418 |
|
|
\[ {}2 y^{\prime \prime }+2 y^{\prime }+3 y = 0 \] \(r = -{\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.419 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.543 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.593 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.192 |
|
|
\[ {}4 y^{\prime \prime }+20 y^{\prime }+25 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+3 y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.168 |
|
|
\[ {}y^{\prime \prime } = 4 y \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.666 |
|
|
\[ {}4 y^{\prime \prime }-8 y^{\prime }+7 y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.282 |
|
|
\[ {}2 y^{\prime \prime }+y^{\prime }-y = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.425 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.649 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.579 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.42 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+5 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.832 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.096 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.019 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+2 y = 0 \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.768 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }-9 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.98 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+10 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.615 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+10 x y^{\prime }+8 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.474 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.625 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.76 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.393 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.582 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+3 y = 0 \] \(r = -\frac {3}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.976 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {7}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.53 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-16 y = 0 \] \(r = \frac {63}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.069 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-10 y = 6 \,{\mathrm e}^{4 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 \sin \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.191 |
|
|
\[ {}y^{\prime \prime }+10 y^{\prime }+25 y = 14 \,{\mathrm e}^{-5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.946 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 25 x^{2}+12 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.792 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 20 \,{\mathrm e}^{-2 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 14 \sin \left (2 x \right )-18 \cos \left (2 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.423 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.309 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 12 x -10 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.402 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 6 \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.877 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.22 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 10 x^{4}+2 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.957 |
|
|
\[ {}y^{\prime \prime }+4 y = 4 \cos \left (2 x \right )+6 \cos \left (x \right )+8 x^{2}-4 x \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.109 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \sin \left (3 x \right )+4 \sin \left (x \right )-26 \,{\mathrm e}^{-2 x}+27 x^{3} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
9.024 |
|
|
\[ {}y^{\prime \prime }-3 y = {\mathrm e}^{2 x} \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.917 |
|
|
\[ {}y^{\prime \prime }+4 y = \tan \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.411 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.049 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 64 x \,{\mathrm e}^{-x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.817 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = {\mathrm e}^{-x} \sec \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.585 |
|
|
\[ {}2 y^{\prime \prime }+3 y^{\prime }+y = {\mathrm e}^{-3 x} \] \(r = {\frac {1}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \frac {1}{1+{\mathrm e}^{-x}} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.905 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.451 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.959 |
|
|
\[ {}y^{\prime \prime }+y = x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.374 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.986 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.848 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.872 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = {\mathrm e}^{-x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.718 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = \left (x^{2}-1\right )^{2} \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.388 |
|
|
\[ {}\left (x^{2}+x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-\left (2+x \right ) y = x \left (1+x \right )^{2} \] \(r = \frac {x^{2}+4 x +6}{4 \left (1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.458 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.786 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = {\mathrm e}^{2 x} x^{2} \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.931 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.461 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.846 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.528 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+6 y = 0 \] \(r = -{\frac {23}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.313 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-5 y = x \] \(r = 6\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.971 |
|
|
\[ {}y^{\prime \prime }-y = {\mathrm e}^{3 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime \prime }+9 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
14.71 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+4 y = x \] \(r = -{\frac {15}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.164 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = {\mathrm e}^{x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.602 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+4 y = \sin \left (x \right ) \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.681 |
|
|
\[ {}y^{\prime \prime }+y = {\mathrm e}^{-x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.026 |
|
|
\[ {}y^{\prime \prime }-y = \cos \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.87 |
|
|
\[ {}y^{\prime \prime } = \tan \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
20.485 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = \ln \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
9.974 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 2 x -1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.724 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.707 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = \cos \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.863 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-y = x \,{\mathrm e}^{x} \sin \left (x \right ) \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.601 |
|
|
\[ {}y^{\prime \prime }+9 y = \sec \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.189 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = x \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.442 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {2}{x} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.598 |
|
|
\[ {}y^{\prime \prime }+4 y = \tan \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.031 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
6.595 |
|
|
\[ {}y^{\prime \prime }+9 y = -3 \cos \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.618 |
|
|
\[ {}y^{\prime \prime } = -3 y \] \(r = -3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
13.961 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.12 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.589 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.213 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.608 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.445 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.565 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.976 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] \(r = \frac {-4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.591 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.682 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.8 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] \(r = \frac {-x^{4}+12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.677 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (16 x^{2}+3\right ) y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.508 |
|
|
\[ {}t y^{\prime \prime }-y^{\prime } = 2 t^{2} \] \(r = \frac {3}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.659 |
|
|
\[ {}2 y^{\prime \prime }+t y^{\prime }-2 y = 10 \] \(r = \frac {t^{2}}{16}+\frac {5}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.484 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }+x^{5} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.078 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime }+x = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.556 |
|
|
\[ {}y^{\prime \prime }+\beta ^{2} y = 0 \] \(r = -\beta ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.576 |
|
|
\[ {}x^{3} y^{\prime \prime }-x^{2} y^{\prime } = -x^{2}+3 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.612 |
|
|
\[ {}y^{\prime \prime }+y = -\cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.909 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.773 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x^{2}+2 x +1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.52 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x y^{\prime }+2 y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.355 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+2 y = 0 \] \(r = -\frac {2}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.414 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }-2 y = 0 \] \(r = \frac {21}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.201 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.175 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.403 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.388 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+5 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.615 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \,{\mathrm e}^{2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.607 |
|
|
\[ {}y^{\prime \prime }+16 y = 4 \cos \left (x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 9 x^{2}+4 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.883 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.323 |
|
|
\[ {}5 y^{\prime \prime }+2 y^{\prime }+4 y = 0 \] \(r = -{\frac {19}{25}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.092 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+4 y = 1 \] \(r = -{\frac {15}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.799 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+4 y = \sin \left (x \right ) \] \(r = -{\frac {15}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.089 |
|
|
\[ {}t y^{\prime \prime }+4 y^{\prime } = t^{2} \] \(r = \frac {2}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.355 |
|
|
\[ {}\left (t^{2}+9\right ) y^{\prime \prime }+2 t y^{\prime } = 0 \] \(r = \frac {9}{\left (t^{2}+9\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.044 |
|
|
\[ {}t^{2} y^{\prime \prime }-3 t y^{\prime }+5 y = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.004 |
|
|
\[ {}t y^{\prime \prime }+y^{\prime } = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.972 |
|
|
\[ {}t^{2} y^{\prime \prime }-2 y^{\prime } = 0 \] \(r = \frac {1+2 t}{t^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 3\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.863 |
|
|
\[ {}t y^{\prime \prime }-y^{\prime }+4 t^{3} y = 0 \] \(r = \frac {-16 t^{4}+3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.806 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.726 |
|
|
\[ {}y^{\prime \prime } = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.167 |
|
|
\[ {}y^{\prime \prime } = f \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.664 |
|
|
\[ {}y^{\prime \prime } = k \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime } = 4 \sin \left (x \right )-4 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
2.115 |
|
|
\[ {}z^{\prime \prime }+3 z^{\prime }+2 z = 24 \,{\mathrm e}^{-3 t}-24 \,{\mathrm e}^{-4 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.63 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.881 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.337 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-2 x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-3 x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2}-x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.596 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3}+2 = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.592 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{4}-6 = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{5}+24 = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{2} = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.434 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x^{3} = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.037 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y-x = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.172 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{2} y-x^{3}-\frac {1}{x} = 0 \] \(r = \frac {4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.931 |
|
|
\[ {}y^{\prime \prime }+c y^{\prime }+k y = 0 \] \(r = \frac {c^{2}}{4}-k\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.468 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.956 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.727 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.941 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.873 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.921 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.786 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.717 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.818 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.169 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.121 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = 1 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.953 |
|
|
\[ {}x^{4} y^{\prime \prime }+x^{3} y^{\prime }-4 x^{2} y = x \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.736 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.913 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 8 \sqrt {x}\, \left (1+\ln \left (x \right )\right ) \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.844 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.413 |
|
|
\[ {}y^{\prime \prime }+20 y^{\prime }+500 y = 100000 \cos \left (100 x \right ) \] \(r = -400\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.781 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.978 |
|
|
\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }-y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.969 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.065 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.559 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 6 x^{3} {\mathrm e}^{x} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.158 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-24 y = 16-\left (2+x \right ) {\mathrm e}^{4 x} \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.036 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.185 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.064 |
|
|
\[ {}a y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.092 |
|
|
\[ {}y^{\prime \prime } = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.986 |
|
|
\[ {}y^{\prime \prime } = x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.036 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.35 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.96 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.497 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 1 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.868 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.882 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 1+x \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.089 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2}+x +1 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.2 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{3}+x^{2}+x +1 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.204 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \sin \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.121 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \cos \left (x \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.461 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.809 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.117 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 1+x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.456 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}+x +1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.502 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{3}+x^{2}+x +1 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.882 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.535 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \cos \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.253 |
|
|
\[ {}y^{\prime \prime }+y = 1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.953 |
|
|
\[ {}y^{\prime \prime }+y = x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+y = 1+x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+y = x^{2}+x +1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime \prime }+y = x^{3}+x^{2}+x +1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.698 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.876 |
|
|
\[ {}y^{\prime \prime }-\frac {2 y}{x^{2}} = x \,{\mathrm e}^{-\sqrt {x}} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.401 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = x \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.743 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0 \] \(r = -\frac {a^{2}}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.368 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-c^{2} y = 0 \] \(r = \frac {-4 c^{2} x^{2}+4 c^{2}-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.397 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 2 x^{3}-x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.483 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 8 x^{3} \sin \left (x \right )^{2} \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.69 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5} \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.961 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = x^{1+m} \] \(r = \frac {x \left (x^{3}-8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.704 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.753 |
|
|
\[ {}\cos \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+y \cos \left (x \right )^{2} = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y = -3 \,{\mathrm e}^{x^{2}} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.708 |
|
|
\[ {}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x \] \(r = -b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.88 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.444 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = {\mathrm e}^{x^{2}} \sec \left (x \right ) \] \(r = -6\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.422 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 \left (x^{2}+1\right ) y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.449 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0 \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.969 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.171 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
1.082 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.481 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
1.222 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] \(r = \frac {15}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
1.273 |
|
|
\[ {}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] \(r = \frac {-x^{2}-234}{4 \left (x^{2}+3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.48 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] \(r = \frac {8}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
1.233 |
|
|
\[ {}3 y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {x^{2}}{36}+\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.226 |
|
|
\[ {}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0 \] \(r = \frac {x^{2}}{25}-\frac {11}{5}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.582 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0 \] \(r = \frac {x \left (x^{3}+8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.26 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.989 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.579 |
|
|
\[ {}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0 \] \(r = -\frac {8}{\left (x^{2}-6 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.768 |
|
|
\[ {}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0 \] \(r = \frac {15 x^{2}+90 x -27}{4 \left (x^{2}+6 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.381 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{2} y = 0 \] \(r = \frac {t^{4}-4 t^{3}+3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.844 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.886 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
1.096 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.308 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.799 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.897 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.003 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = -\frac {8}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.075 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.171 |
|
|
\[ {}2 y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}}{16}-\frac {5}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.338 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.732 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.928 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.746 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {11 x^{2}-24}{4 \left (x^{2}-4\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.282 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
|
|||||
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.933 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.815 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.761 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.438 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.677 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.876 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.85 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.971 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.032 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.9 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.901 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] \(r = \frac {81 x^{2}-108 x +54}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.651 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}-12 x -20}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.633 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] \(r = \frac {-x +36}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.134 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-\left (4+6 x \right ) y = 0 \] \(r = \frac {24 x^{2}+40 x +15}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.416 |
|
|
\[ {}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0 \] \(r = \frac {3 x^{2}-9}{\left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.817 |
|
|
\[ {}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0 \] \(r = \frac {2 x^{4}-5 x^{2}+3}{\left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.011 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.106 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.681 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0 \] \(r = -\frac {24}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.13 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y = 0 \] \(r = \frac {8}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.868 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}+6}{4 \left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.03 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y = 0 \] \(r = \frac {-x^{2}+6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.553 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y = 0 \] \(r = \frac {2 x^{2}-33}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.367 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y = 0 \] \(r = \frac {165 x^{2}+6}{4 \left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.878 |
|
|
\[ {}\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (-2+x \right ) y^{\prime }+36 y = 0 \] \(r = \frac {8 x^{2}-32 x -100}{\left (2 x^{2}-8 x +11\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y = 0 \] \(r = -\frac {1}{4}+\frac {1}{4} x^{2}-\frac {3}{2} x\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.477 |
|
|
\[ {}\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y = 0 \] \(r = \frac {48}{\left (x^{2}-8 x +14\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.839 |
|
|
\[ {}\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y = 0 \] \(r = -\frac {210}{\left (2 x^{2}+4 x +5\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 x y = 0 \] \(r = -\frac {x \left (x^{3}+8\right )}{4 \left (x^{3}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.331 |
|
|
\[ {}\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y = 0 \] \(r = \frac {3 x^{3} \left (5 x^{5}+6\right )}{\left (2 x^{5}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
83.076 |
|
|
\[ {}y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y = 0 \] \(r = \frac {x^{5} \left (x^{7}-16\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -12\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y = 0 \] \(r = -\frac {128 x^{6}}{\left (x^{8}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 10\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
160.967 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 y x^{4} = 0 \] \(r = \frac {x^{4} \left (x^{6}-14\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -10\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.905 |
|
|
\[ {}\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-24 x -6}{4 \left (1+3 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
55.138 |
|
|
\[ {}\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y = 0 \] \(r = \frac {-9 x^{2}-12 x -18}{4 \left (3 x^{2}+x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.248 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}-10 x -21}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.903 |
|
|
\[ {}\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-4 x -24}{4 \left (x +4\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.878 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y = 0 \] \(r = \frac {-x^{2}+6 x +10}{\left (2 x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y = 0 \] \(r = \frac {3 x^{2}-60 x +36}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}-2 x +5}{4 \left (2 x^{2}+x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.07 |
|
|
\[ {}\left (x +3\right ) y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }-\left (2-x \right ) y = 0 \] \(r = \frac {35}{4 \left (x +3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y = 0 \] \(r = \frac {x^{2}}{4}-\frac {13}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] \(r = \frac {9 x^{2}}{16}-\frac {3}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.428 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.333 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] \(r = \frac {21 x^{4}+18 x^{3}+27 x^{2}-2 x -3}{16 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.671 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y = 0 \] \(r = \frac {4 x^{4}-4 x^{3}+15 x^{2}-4 x -2}{9 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.122 |
|
|
\[ {}12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y = 0 \] \(r = \frac {9 x^{4}-30 x^{3}-197 x^{2}-190 x -95}{576 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.388 |
|
|
\[ {}x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y = 0 \] \(r = \frac {-96 x^{4}-16 x^{3}-97 x^{2}-12 x -24}{4 \left (10 x^{3}+x^{2}+5 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.028 |
|
|
\[ {}18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y = 0 \] \(r = \frac {x^{4}-18 x^{3}-45 x^{2}-18 x -27}{144 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.391 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y = 0 \] \(r = \frac {4 x^{2}+4 x +5}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-14 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.968 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}+2 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.091 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (1-2 x \right ) y = 0 \] \(r = \frac {-3+16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] \(r = \frac {x^{2}+38 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.275 |
|
|
\[ {}2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-30 x -35}{16 \left (x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.286 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] \(r = \frac {3 x^{2}-6 x -7}{4 \left (x^{2}+4 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.541 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3-8 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.772 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{2}+4 x +28}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.209 |
|
|
\[ {}x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y = 0 \] \(r = \frac {48 x^{2}+8 x +91}{4 \left (4 x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.213 |
|
|
\[ {}2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y = 0 \] \(r = -\frac {35}{16 \left (3 x +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.952 |
|
|
\[ {}x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y = 0 \] \(r = \frac {3 x^{2}-126 x +21}{4 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.171 |
|
|
\[ {}8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] \(r = \frac {-7 x^{4}-26 x^{2}-15}{64 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.553 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y = 0 \] \(r = \frac {-x^{2}+2}{\left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.059 |
|
|
\[ {}x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 x y = 0 \] \(r = \frac {35 x^{4}+74 x^{2}-8}{4 \left (x^{3}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.099 |
|
|
\[ {}4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y = 0 \] \(r = \frac {-15 x^{4}-42 x^{2}+9}{64 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.785 |
|
|
\[ {}3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y = 0 \] \(r = \frac {-5 x^{4}-4 x^{2}-35}{36 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.487 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y = 0 \] \(r = \frac {-3 x^{4}-72 x^{2}+128}{16 \left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.221 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y = 0 \] \(r = \frac {-3 x^{2}+24}{16 \left (x^{2}+2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.164 |
|
|
\[ {}2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y = 0 \] \(r = \frac {20 x^{4}+12 x^{2}+21}{16 \left (2 x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.838 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 x y = 0 \] \(r = \frac {3 x^{4}+14 x^{2}+8}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.987 |
|
|
\[ {}2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y = 0 \] \(r = \frac {36 x^{2}+21}{16 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.195 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{2}-5}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.014 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y = 0 \] \(r = \frac {36 x^{4}-132 x^{2}-35}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.312 |
|
|
\[ {}9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] \(r = \frac {-9 x^{4}+6 x^{2}-5}{36 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.21 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] \(r = \frac {-x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.978 |
|
|
\[ {}8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y = 0 \] \(r = \frac {132 x^{4}+148 x^{2}-7}{64 \left (2 x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.398 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.856 |
|
|
\[ {}6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y = 0 \] \(r = -\frac {35}{144 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.779 |
|
|
\[ {}28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y = 0 \] \(r = \frac {33}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.919 |
|
|
\[ {}8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y = 0 \] \(r = -\frac {7}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.798 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.713 |
|
|
\[ {}3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] \(r = -\frac {5}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.095 |
|
|
\[ {}4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y = 0 \] \(r = -\frac {7}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.834 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] \(r = \frac {7 x^{2}+10 x -1}{4 x^{2} \left (-1+x \right )^{4}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2, 4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.058 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-24 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.957 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] \(r = \frac {3 x^{2}-1}{\left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.126 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] \(r = \frac {x^{2}+6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.976 |
|
|
\[ {}4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y = 0 \] \(r = \frac {2 x^{2}-4 x -1}{4 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.408 |
|
|
\[ {}x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y = 0 \] \(r = \frac {10 x^{2}-8 x -1}{4 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.25 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y = 0 \] \(r = \frac {4 x^{4}-12 x^{3}+33 x^{2}-18 x -9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.306 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] \(r = \frac {9 x^{4}-12 x^{3}-16 x^{2}-4 x -1}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.517 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y = 0 \] \(r = \frac {4 x^{4}+4 x^{3}-31 x^{2}-8 x -16}{64 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.305 |
|
|
\[ {}9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y = 0 \] \(r = \frac {x^{4}+6 x^{3}+3 x^{2}-18 x -9}{36 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.428 |
|
|
\[ {}36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y = 0 \] \(r = \frac {-32 x^{2}+48 x -9}{36 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.198 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.917 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] \(r = \frac {8 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.827 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y = 0 \] \(r = \frac {5 x^{2}+8 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.959 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y = 0 \] \(r = \frac {5 x^{2}-20 x -4}{16 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.016 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (3 x +4\right ) y = 0 \] \(r = \frac {21 x^{2}+6 x -1}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] \(r = \frac {32 x^{2}+56 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y = 0 \] \(r = \frac {-x^{2}+82 x -1}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.932 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-4 x^{2}-1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.762 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y = 0 \] \(r = \frac {3 x^{4}-10 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.939 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{2}-4}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y = 0 \] \(r = \frac {2 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] \(r = \frac {-3 x^{4}-16}{16 \left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.242 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y = 0 \] \(r = \frac {-6 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.904 |
|
|
\[ {}4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] \(r = \frac {153 x^{4}+704 x^{2}-256}{64 \left (x^{3}+4 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.405 |
|
|
\[ {}3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] \(r = \frac {-5 x^{4}+18 x^{2}-81}{36 \left (x^{3}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.498 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y = 0 \] \(r = \frac {4 x^{4}-24 x^{2}-9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.884 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-4 x^{2}-1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.608 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+4 x \right ) y = 0 \] \(r = \frac {32 x^{2}+16 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.82 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.739 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] \(r = \frac {15 x^{2}-6 x -1}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y = 0 \] \(r = \frac {-x^{4}-98 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.046 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-36 x^{2}-9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.293 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 x y = 0 \] \(r = \frac {35 x^{4}+22 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.998 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-40 x^{2}-4}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.523 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.135 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.63 |
|
|
\[ {}16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.115 |
|
|
\[ {}x^{2} \left (3 x +4\right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.621 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.708 |
|
|
\[ {}x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.947 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y = 0 \] \(r = \frac {-3 x^{2}-32 x +128}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.04 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \] \(r = \frac {21 x^{2}-20 x +24}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.049 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y = 0 \] \(r = \frac {24 x^{4}+66 x^{2}+63}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.158 |
|
|
\[ {}x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y = 0 \] \(r = \frac {5 x^{4}-68 x^{2}+35}{4 \left (2 x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.266 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] \(r = \frac {3 x +4}{4 x \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y = 0 \] \(r = \frac {-27 x -48}{16 x \left (3 x +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y = 0 \] \(r = \frac {-x^{2}-8 x +8}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.006 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 x y = 0 \] \(r = \frac {-48 x +15}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.981 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y = 0 \] \(r = \frac {x^{2}+14 x +35}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.857 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y = 0 \] \(r = \frac {77 x^{2}+86 x +35}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.968 |
|
|
\[ {}4 x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y = 0 \] \(r = \frac {33 x^{2}+132 x +60}{16 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.865 |
|
|
\[ {}3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y = 0 \] \(r = \frac {7 x^{2}+450 x +1215}{36 \left (x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.042 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y = 0 \] \(r = \frac {80 x^{2}-28 x +35}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.97 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y = 0 \] \(r = \frac {-x^{2}+20 x +24}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.913 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y = 0 \] \(r = \frac {24 x^{2}-16 x +6}{\left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.869 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y = 0 \] \(r = \frac {15 x^{2}+4 x +24}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.964 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y = 0 \] \(r = \frac {35 x^{2}+80 x +48}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.96 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y = 0 \] \(r = \frac {-30 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.981 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y = 0 \] \(r = \frac {x^{4}-12 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.794 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 x^{2}+1\right ) y = 0 \] \(r = \frac {4 x^{4}-32 x^{2}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.093 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y = 0 \] \(r = \frac {4 x^{4}+24 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y = 0 \] \(r = \frac {9 x^{4}-60 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.474 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y = 0 \] \(r = \frac {3 x^{4}+6 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y = 0 \] \(r = \frac {-x^{4}+22 x^{2}+35}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.075 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y = 0 \] \(r = \frac {99 x^{4}+150 x^{2}+63}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.079 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y = 0 \] \(r = \frac {78 x^{2}+99}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.073 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y = 0 \] \(r = \frac {10 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] \(r = \frac {6}{\left (t^{2}+2 t -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.202 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.454 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = \frac {2 t^{2}-3}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.825 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] \(r = \frac {6 t^{2}-7}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.751 |
|
|
\[ {}\left (1+2 t \right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y = 0 \] \(r = \frac {4 t^{2}+2}{\left (1+2 t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.827 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.658 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = 0 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.394 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] \(r = \frac {21}{4}+6 t +\frac {1}{4} t^{4}+t^{3}+\frac {3}{2} t^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.023 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] \(r = \frac {4 t^{2}+4 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.87 |
|
|
\[ {}2 t y^{\prime \prime }+\left (t +1\right ) y^{\prime }-2 y = 0 \] \(r = \frac {t^{2}+18 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.213 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (t +1\right ) y = 0 \] \(r = \frac {-3-8 t}{16 t^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.829 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) y^{\prime }-y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+t y = 0 \] \(r = \frac {t^{4}-2 t^{2}+8}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[_Lienard] |
✓ |
✓ |
0.854 |
|
|
\[ {}t^{2} y^{\prime \prime }+t \left (t +1\right ) y^{\prime }-y = 0 \] \(r = \frac {t^{2}+2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.613 |
|
|
\[ {}t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y = 0 \] \(r = \frac {t^{2}+24}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.779 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (t^{2}-3 t \right ) y^{\prime }+3 y = 0 \] \(r = \frac {t^{2}-6 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.684 |
|
|
\[ {}t y^{\prime \prime }+t y^{\prime }+2 y = 0 \] \(r = \frac {t -8}{4 t}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0 \] \(r = \frac {t^{4}-20 t^{2}-1}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.42 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}+2 t -1}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.639 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y = 0 \] \(r = \frac {4 \lambda \,z^{2}+3 z^{2}-4 \lambda -6}{4 \left (z^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
1.431 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] \(r = \frac {z^{2}+2 z -3}{16 z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.865 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] \(r = z^{2}-2 z -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.712 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+y z = 0 \] \(r = \frac {-z^{2}+2}{z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.855 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] \(r = \frac {4 z^{2}-12 z -1}{4 z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] \(r = x^{2}-3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_erf] |
✓ |
✓ |
0.53 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.617 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0 \] \(r = \frac {x \left (x^{3}+8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.744 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] \(r = \frac {-4 x^{2}+6}{\left (4 x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.931 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] \(r = \frac {15}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {1}{4} x^{2}-x -\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.795 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}+6}{4 \left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.625 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}}{64}-\frac {7}{8}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Lienard] |
✓ |
✓ |
0.776 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {9}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.219 |
|
|
|
|||||
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x -32}{64 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] \(r = \frac {324 x^{2}-252 x -35}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.01 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] \(r = \frac {x^{2}+16 x +40}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.032 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] \(r = \frac {4 x^{2}-60 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.736 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (2 x +1\right ) y = 0 \] \(r = \frac {x^{2}-2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y = 0 \] \(r = \frac {4 x^{2}-4 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.804 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] \(r = \frac {x^{2}+10 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}-6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.717 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.027 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] \(r = \frac {x^{2}-2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.683 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{2}+4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] \(r = \frac {4 x^{2}-4 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.746 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] \(r = \frac {-x^{2}+8 x +8}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.754 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {7}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.144 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}+6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.632 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] \(r = \frac {x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.63 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y = 0 \] \(r = \frac {x^{2}+12 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}-10 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.818 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.757 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] \(r = \frac {2+x}{x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] \(r = \frac {x^{2}+2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.699 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.611 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.933 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.513 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y = 0 \] \(r = \frac {100 x^{2}-60 x +5}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] \(r = \frac {n^{2}-2 n x +x^{2}-2 n -4 x}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = \frac {-10 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.121 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0 \] \(r = \frac {4 x^{2}+4 x +15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] \(r = \frac {-12 x^{4}+156 x^{3}+297 x^{2}-78 x -3}{16 \left (2 x^{3}-7 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.224 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.664 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.635 |
|
|
\[ {}x^{2} \left (1-4 x \right ) y^{\prime \prime }-\frac {x y^{\prime }}{2}-\frac {3 x y}{4} = 0 \] \(r = \frac {-48 x^{2}-20 x +5}{16 \left (4 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.466 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] \(r = \frac {x^{2}-2 x +35}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] \(r = \frac {x^{2}-10 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}+8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.636 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = 0 \] \(r = \frac {5 x^{2}+36 x +4}{16 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.131 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0 \] \(r = \frac {-4 x^{2}+4 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.81 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {-3 x +8}{16 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.524 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \] \(r = \frac {-3 x^{2}+66 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.823 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \] \(r = \frac {72 x^{2}-72 x -5}{36 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.862 |
|
|
\[ {}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0 \] \(r = \frac {3 x^{2}-6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.853 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.586 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.611 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] \(r = \frac {-a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {a^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {-n^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.171 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] \(r = \frac {2 a^{2}-x^{2}}{x^{2} a^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.944 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.056 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] \(r = \frac {q^{2} x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.771 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.935 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.618 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.356 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.335 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-8 x +18}{4 \left (2 x -3\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.19 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.694 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.486 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.73 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = \frac {4 x^{2}-4 x -3}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.51 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.718 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.901 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {x +8}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.488 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{\left (-1+x \right )^{2} x}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.678 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+x^{2} y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.58 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] \(r = \frac {x^{6}+2 x^{5}-5 x^{4}-16 x^{3}+24 x^{2}+24 x +12}{4 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.403 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (2 x +1\right ) \left (-y+x y^{\prime }\right ) = 0 \] \(r = \frac {-4 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.84 |
|
|
\[ {}2 \left (2-x \right ) x^{2} y^{\prime \prime }-x \left (4-x \right ) y^{\prime }+\left (-x +3\right ) y = 0 \] \(r = -\frac {3}{16 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.75 |
|
|
\[ {}\left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] \(r = \frac {4-x}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.553 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.735 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.192 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.169 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] \(r = \frac {12 x^{2}-13}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.891 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+4 x -1}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}+30 x -3}{4 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.777 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.794 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.391 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}-32 x +180}{4 \left (x^{2}-2 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.313 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}-32 x +180}{4 \left (x^{2}-2 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.474 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.445 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}+4 x +12}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] \(r = \frac {6}{x^{2}+1}\) \(L = [1, 4, 6, 12]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.411 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] \(r = \frac {7 x^{2}+20}{4 \left (x^{2}+2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.173 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.607 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.167 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] \(r = \frac {9 x^{2}+30 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.148 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.551 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] \(r = \frac {4 x^{2}+4 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
1.04 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.526 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}+48}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.853 |
|
|
\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \] \(r = -\frac {\lambda }{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.665 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.804 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.67 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] \(r = \frac {-4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.95 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.599 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] \(r = \frac {-x^{4}+12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.003 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0 \] \(r = \frac {x \left (x^{3}-8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.78 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.475 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {2 x^{2}-3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.717 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.358 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y = 0 \] \(r = \frac {30 x^{2}-31}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.881 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.433 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.645 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {-3 x^{2}+18 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.709 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.647 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.639 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {48 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[_Jacobi] |
✓ |
✓ |
0.888 |
|
|
\[ {}4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = \frac {-4 t^{2}+20 t -19}{16 \left (t^{2}-3 t +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.971 |
|
|
\[ {}2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y = 0 \] \(r = \frac {60 t^{2}-308 t +381}{16 \left (t^{2}-5 t +6\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.975 |
|
|
\[ {}3 t \left (t +1\right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {7 t +12}{36 t \left (t +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.002 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4} = 0 \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.638 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4} = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.658 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.685 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.73 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] \(r = -\frac {2}{x^{2} \left (x^{2}-1\right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.603 |
|
|
\[ {}2 x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}+4 x +12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.433 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.653 |
|
|
\[ {}u^{\prime \prime }+\frac {u}{x^{2}} = 0 \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.557 |
|
|
\[ {}u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] \(r = -\frac {2}{\left (1+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.885 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.585 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] \(r = \frac {2}{\left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{2 x^{4}} = 0 \] \(r = -\frac {1}{2 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.695 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.451 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.466 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.434 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-x y = 0 \] \(r = \frac {8 x -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.706 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {9 x^{2}+12 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] \(r = \frac {21 x^{4}+18 x^{3}+27 x^{2}-2 x -3}{16 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.388 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] \(r = \frac {7 x^{2}+10 x -1}{4 x^{2} \left (-1+x \right )^{4}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2, 4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-24 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.927 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.494 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.506 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] \(r = \frac {x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.475 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 y x^{4} = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.501 |
|
|
\[ {}x^{3} y^{\prime \prime }+y^{\prime }-\frac {y}{x} = 0 \] \(r = \frac {-2 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.482 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.496 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.757 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.596 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2+x \right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}+2}{4 \left (1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.8 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.644 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {2 x^{2}-3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.671 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.48 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] \(r = \frac {15}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.718 |
|
|
\[ {}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] \(r = \frac {-x^{2}-234}{4 \left (x^{2}+3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.876 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] \(r = \frac {8}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.757 |
|
|
\[ {}3 y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {x^{2}}{36}+\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.164 |
|
|
\[ {}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0 \] \(r = \frac {x^{2}}{25}-\frac {11}{5}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0 \] \(r = \frac {x \left (x^{3}+8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0 \] \(r = -\frac {8}{\left (x^{2}-6 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.936 |
|
|
\[ {}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0 \] \(r = \frac {15 x^{2}+90 x -27}{4 \left (x^{2}+6 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.859 |
|
|
\[ {}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{3} y = 0 \] \(r = \frac {-3 t^{4}+3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.649 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y = 0 \] \(r = \frac {16 x -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.55 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.47 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.606 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.747 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.481 |
|
|
\[ {}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.548 |
|
|
\[ {}\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {t^{2}-4 t +6}{4 \left (-1+t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = -\frac {8}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.601 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}2 y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}}{16}-\frac {5}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.771 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.436 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.614 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.434 |
|
|
\[ {}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {11 x^{2}-24}{4 \left (x^{2}-4\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.575 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.686 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.664 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.121 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.758 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.405 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (-1+x \right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}\left (1-2 x \right ) y^{\prime \prime }+2 y^{\prime }+\left (2 x -3\right ) y = 0 \] \(r = \frac {4 x^{2}-8 x +6}{\left (2 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+4 x \right ) y^{\prime }+\left (2 x +1\right ) y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.699 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.644 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.367 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.733 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.48 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}\left (2 x +1\right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (1+x \right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.986 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.81 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.553 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] \(r = \frac {81 x^{2}-108 x +54}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.006 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.547 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.365 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.664 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.534 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.484 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.641 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.619 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.728 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.64 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.634 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] \(r = \frac {81 x^{2}-108 x +54}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.748 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}-12 x -20}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}\left (1-x \right ) x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] \(r = \frac {-x +36}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.669 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-\left (4+6 x \right ) y = 0 \] \(r = \frac {24 x^{2}+40 x +15}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.896 |
|
|
\[ {}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0 \] \(r = \frac {3 x^{2}-9}{\left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.107 |
|
|
\[ {}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0 \] \(r = \frac {2 x^{4}-5 x^{2}+3}{\left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.308 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.583 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.415 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0 \] \(r = -\frac {24}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.62 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y = 0 \] \(r = \frac {8}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.675 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}+6}{4 \left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.588 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y = 0 \] \(r = \frac {-x^{2}+6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.874 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y = 0 \] \(r = \frac {2 x^{2}-33}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.665 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.463 |
|
|
\[ {}\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (-2+x \right ) y^{\prime }+36 y = 0 \] \(r = \frac {8 x^{2}-32 x -100}{\left (2 x^{2}-8 x +11\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.439 |
|
|
\[ {}y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y = 0 \] \(r = -\frac {1}{4}+\frac {1}{4} x^{2}-\frac {3}{2} x\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.545 |
|
|
\[ {}\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y = 0 \] \(r = \frac {48}{\left (x^{2}-8 x +14\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.144 |
|
|
\[ {}\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y = 0 \] \(r = -\frac {210}{\left (2 x^{2}+4 x +5\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.365 |
|
|
\[ {}\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 x y = 0 \] \(r = -\frac {x \left (x^{3}+8\right )}{4 \left (x^{3}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.381 |
|
|
\[ {}\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y = 0 \] \(r = \frac {3 x^{3} \left (5 x^{5}+6\right )}{\left (2 x^{5}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
62.585 |
|
|
\[ {}y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y = 0 \] \(r = \frac {x^{5} \left (x^{7}-16\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -12\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.787 |
|
|
\[ {}\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y = 0 \] \(r = -\frac {128 x^{6}}{\left (x^{8}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 10\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
168.78 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 y x^{4} = 0 \] \(r = \frac {x^{4} \left (x^{6}-14\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -10\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.809 |
|
|
\[ {}\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-24 x -6}{4 \left (1+3 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.861 |
|
|
\[ {}\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y = 0 \] \(r = \frac {-9 x^{2}-12 x -18}{4 \left (3 x^{2}+x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.066 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}-10 x -21}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.845 |
|
|
\[ {}\left (x +4\right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-4 x -24}{4 \left (x +4\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.852 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y = 0 \] \(r = \frac {-x^{2}+6 x +10}{\left (2 x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.967 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y = 0 \] \(r = \frac {3 x^{2}-60 x +36}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}-2 x +5}{4 \left (2 x^{2}+x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.882 |
|
|
\[ {}\left (x +3\right ) y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }-\left (2-x \right ) y = 0 \] \(r = \frac {35}{4 \left (x +3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.746 |
|
|
\[ {}y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.612 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y = 0 \] \(r = \frac {x^{2}}{4}-\frac {13}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.651 |
|
|
\[ {}2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y = 0 \] \(r = \frac {9 x^{2}}{16}-\frac {3}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.372 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] \(r = \frac {21 x^{4}+18 x^{3}+27 x^{2}-2 x -3}{16 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.336 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y = 0 \] \(r = \frac {4 x^{4}-4 x^{3}+15 x^{2}-4 x -2}{9 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.809 |
|
|
\[ {}12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y = 0 \] \(r = \frac {9 x^{4}-30 x^{3}-197 x^{2}-190 x -95}{576 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.236 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+4 y = 0 \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.398 |
|
|
\[ {}18 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y = 0 \] \(r = \frac {x^{4}-18 x^{3}-45 x^{2}-18 x -27}{144 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.234 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y = 0 \] \(r = \frac {4 x^{2}+4 x +5}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.736 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-14 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.787 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}+2 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.697 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (1-2 x \right ) y = 0 \] \(r = \frac {16 x -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.585 |
|
|
\[ {}3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] \(r = \frac {x^{2}+38 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.783 |
|
|
\[ {}2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-30 x -35}{16 \left (x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.981 |
|
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] \(r = \frac {3 x^{2}-6 x -7}{4 \left (x^{2}+4 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.369 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3-8 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{2}+4 x +28}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y = 0 \] \(r = \frac {48 x^{2}+8 x +91}{4 \left (4 x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.058 |
|
|
\[ {}2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y = 0 \] \(r = -\frac {35}{16 \left (3 x +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.041 |
|
|
\[ {}x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y = 0 \] \(r = \frac {3 x^{2}-126 x +21}{4 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.284 |
|
|
\[ {}8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] \(r = \frac {-7 x^{4}-26 x^{2}-15}{64 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.203 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y = 0 \] \(r = \frac {-x^{2}+2}{\left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 x y = 0 \] \(r = \frac {35 x^{4}+74 x^{2}-8}{4 \left (x^{3}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.003 |
|
|
\[ {}4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y = 0 \] \(r = \frac {-15 x^{4}-42 x^{2}+9}{64 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.203 |
|
|
\[ {}3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y = 0 \] \(r = \frac {-5 x^{4}-4 x^{2}-35}{36 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.989 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y = 0 \] \(r = \frac {-3 x^{4}-72 x^{2}+128}{16 \left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.073 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y = 0 \] \(r = \frac {-3 x^{2}+24}{16 \left (x^{2}+2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.945 |
|
|
\[ {}2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y = 0 \] \(r = \frac {20 x^{4}+12 x^{2}+21}{16 \left (2 x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.256 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 x y = 0 \] \(r = \frac {3 x^{4}+14 x^{2}+8}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.991 |
|
|
\[ {}2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y = 0 \] \(r = \frac {36 x^{2}+21}{16 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.977 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{2}-5}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y = 0 \] \(r = \frac {36 x^{4}-132 x^{2}-35}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.816 |
|
|
\[ {}9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] \(r = \frac {-9 x^{4}+6 x^{2}-5}{36 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y = 0 \] \(r = \frac {-x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y = 0 \] \(r = \frac {132 x^{4}+148 x^{2}-7}{64 \left (2 x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.797 |
|
|
\[ {}6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y = 0 \] \(r = -\frac {35}{144 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.745 |
|
|
\[ {}28 x^{2} \left (1-3 x \right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y = 0 \] \(r = \frac {33}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.81 |
|
|
\[ {}8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y = 0 \] \(r = -\frac {7}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }-4 x \left (-3 x^{2}-3 x +1\right ) y^{\prime }+3 \left (x^{2}-x +1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.698 |
|
|
\[ {}3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] \(r = -\frac {5}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.958 |
|
|
\[ {}4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y = 0 \] \(r = -\frac {7}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] \(r = \frac {7 x^{2}+10 x -1}{4 x^{2} \left (-1+x \right )^{4}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2, 4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.931 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-24 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] \(r = \frac {3 x^{2}-1}{\left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.973 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] \(r = \frac {x^{2}+6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.718 |
|
|
\[ {}4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y = 0 \] \(r = \frac {2 x^{2}-4 x -1}{4 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.217 |
|
|
\[ {}x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y = 0 \] \(r = \frac {10 x^{2}-8 x -1}{4 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.957 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y = 0 \] \(r = \frac {4 x^{4}-12 x^{3}+33 x^{2}-18 x -9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.891 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y = 0 \] \(r = \frac {9 x^{4}-12 x^{3}-16 x^{2}-4 x -1}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.029 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y = 0 \] \(r = \frac {4 x^{4}+4 x^{3}-31 x^{2}-8 x -16}{64 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.876 |
|
|
\[ {}9 x^{2} \left (1+x \right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y = 0 \] \(r = \frac {x^{4}+6 x^{3}+3 x^{2}-18 x -9}{36 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.253 |
|
|
\[ {}36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y = 0 \] \(r = \frac {-32 x^{2}+48 x -9}{36 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.959 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x +3\right ) y^{\prime }+4 y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.903 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y = 0 \] \(r = \frac {8 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (1-x \right ) y = 0 \] \(r = \frac {5 x^{2}+8 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y = 0 \] \(r = \frac {5 x^{2}-20 x -4}{16 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.996 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (3 x +4\right ) y = 0 \] \(r = \frac {21 x^{2}+6 x -1}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5+4 x \right ) y^{\prime }+\left (9+4 x \right ) y = 0 \] \(r = \frac {32 x^{2}+56 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y = 0 \] \(r = \frac {-x^{2}+82 x -1}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.94 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-4 x^{2}-1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.736 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y = 0 \] \(r = \frac {3 x^{4}-10 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.947 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-8 x^{2}-4}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.724 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y = 0 \] \(r = \frac {2 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y = 0 \] \(r = \frac {-3 x^{4}-16}{16 \left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.081 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+\left (2 x^{2}+1\right ) y = 0 \] \(r = \frac {-6 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y = 0 \] \(r = \frac {153 x^{4}+704 x^{2}-256}{64 \left (x^{3}+4 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.057 |
|
|
\[ {}3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y = 0 \] \(r = \frac {-5 x^{4}+18 x^{2}-81}{36 \left (x^{3}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.145 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y = 0 \] \(r = \frac {4 x^{4}-24 x^{2}-9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.795 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-4 x^{2}-1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.642 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+4 x \right ) y = 0 \] \(r = \frac {32 x^{2}+16 x -1}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.816 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.747 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] \(r = \frac {15 x^{2}-6 x -1}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.813 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y = 0 \] \(r = \frac {-x^{4}-98 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.994 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-36 x^{2}-9}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.871 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }-8 x y = 0 \] \(r = \frac {35 x^{4}+22 x^{2}-1}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.961 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y = 0 \] \(r = \frac {x^{4}-40 x^{2}-4}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.857 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.752 |
|
|
\[ {}9 x^{2} \left (x +3\right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.007 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.638 |
|
|
\[ {}16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.902 |
|
|
\[ {}x^{2} \left (3 x +4\right ) y^{\prime \prime }-x \left (4-3 x \right ) y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.629 |
|
|
\[ {}4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.707 |
|
|
\[ {}x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.671 |
|
|
\[ {}9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.884 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y = 0 \] \(r = \frac {-3 x^{2}-32 x +128}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \] \(r = \frac {21 x^{2}-20 x +24}{4 \left (2 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.941 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y = 0 \] \(r = \frac {24 x^{4}+66 x^{2}+63}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.112 |
|
|
\[ {}x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y = 0 \] \(r = \frac {5 x^{4}-68 x^{2}+35}{4 \left (2 x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.098 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }-\left (1+3 x \right ) y = 0 \] \(r = \frac {3 x +4}{4 x \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.862 |
|
|
\[ {}2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y = 0 \] \(r = \frac {-27 x -48}{16 x \left (3 x +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.889 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.71 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y = 0 \] \(r = \frac {-x^{2}-8 x +8}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.989 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 x y = 0 \] \(r = \frac {-48 x +15}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y = 0 \] \(r = \frac {x^{2}+14 x +35}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.728 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y = 0 \] \(r = \frac {77 x^{2}+86 x +35}{4 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.95 |
|
|
\[ {}4 x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y = 0 \] \(r = \frac {33 x^{2}+132 x +60}{16 \left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.854 |
|
|
\[ {}3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y = 0 \] \(r = \frac {7 x^{2}+450 x +1215}{36 \left (x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.0 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y = 0 \] \(r = \frac {80 x^{2}-28 x +35}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y = 0 \] \(r = \frac {-x^{2}+20 x +24}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.917 |
|
|
\[ {}x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y = 0 \] \(r = \frac {24 x^{2}-16 x +6}{\left (2 x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.862 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y = 0 \] \(r = \frac {15 x^{2}+4 x +24}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.967 |
|
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y = 0 \] \(r = \frac {35 x^{2}+80 x +48}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.974 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y = 0 \] \(r = \frac {-30 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.977 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y = 0 \] \(r = \frac {x^{4}-12 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.744 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x^{2}+1\right ) y^{\prime }-\left (-10 x^{2}+1\right ) y = 0 \] \(r = \frac {4 x^{4}-32 x^{2}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y = 0 \] \(r = \frac {4 x^{4}+24 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-3 x^{2}+1\right ) y^{\prime }-4 \left (-3 x^{2}+1\right ) y = 0 \] \(r = \frac {9 x^{4}-60 x^{2}+15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.829 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 x^{2} y = 0 \] \(r = \frac {3 x^{4}+6 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.111 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y = 0 \] \(r = \frac {-x^{4}+22 x^{2}+35}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y = 0 \] \(r = \frac {99 x^{4}+150 x^{2}+63}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y = 0 \] \(r = \frac {78 x^{2}+99}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.041 |
|
|
\[ {}4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y = 0 \] \(r = \frac {10 x^{2}+15}{4 \left (x^{3}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.958 |
|
|
\[ {}y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1} = 0 \] \(r = \frac {6}{\left (t^{2}+2 t -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.045 |
|
|
\[ {}y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = \frac {2 t^{2}-3}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.824 |
|
|
\[ {}\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y = 0 \] \(r = \frac {6 t^{2}-7}{\left (t^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.767 |
|
|
\[ {}\left (1+2 t \right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y = 0 \] \(r = \frac {4 t^{2}+2}{\left (1+2 t \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.76 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1} = 0 \] \(r = -\frac {3}{\left (t^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4+4 t \right ) y = 0 \] \(r = \frac {21}{4}+6 t +\frac {1}{4} t^{4}+t^{3}+\frac {3}{2} t^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-y = 0 \] \(r = \frac {4 t^{2}+4 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.695 |
|
|
\[ {}2 t y^{\prime \prime }+\left (t +1\right ) y^{\prime }-2 y = 0 \] \(r = \frac {t^{2}+18 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.782 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (t +1\right ) y = 0 \] \(r = \frac {-3-8 t}{16 t^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.711 |
|
|
\[ {}2 t^{2} y^{\prime \prime }+\left (t^{2}-t \right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}-2 t -3}{16 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.692 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (-t^{2}+t \right ) y^{\prime }-y = 0 \] \(r = \frac {t^{2}-2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.632 |
|
|
|
|||||
|
\[ {}t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+t y = 0 \] \(r = \frac {t^{4}-2 t^{2}+8}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[_Lienard] |
✓ |
✓ |
0.753 |
|
|
\[ {}t^{2} y^{\prime \prime }+t \left (t +1\right ) y^{\prime }-y = 0 \] \(r = \frac {t^{2}+2 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.619 |
|
|
\[ {}t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y = 0 \] \(r = \frac {t^{2}+24}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.735 |
|
|
\[ {}t^{2} y^{\prime \prime }+\left (t^{2}-3 t \right ) y^{\prime }+3 y = 0 \] \(r = \frac {t^{2}-6 t +3}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}t y^{\prime \prime }+t y^{\prime }+2 y = 0 \] \(r = \frac {t -8}{4 t}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.658 |
|
|
\[ {}t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 t y = 0 \] \(r = \frac {t^{4}-20 t^{2}-1}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.756 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (t +1\right ) y^{\prime }+y = 0 \] \(r = \frac {t^{2}+2 t -1}{4 t^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.606 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+y = 0 \] \(r = \frac {7 z^{2}-10}{4 \left (z^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
1.608 |
|
|
\[ {}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0 \] \(r = \frac {z^{2}+2 z -3}{16 z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0 \] \(r = z^{2}-2 z -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}z y^{\prime \prime }-2 y^{\prime }+y z = 0 \] \(r = \frac {-z^{2}+2}{z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.79 |
|
|
\[ {}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0 \] \(r = \frac {4 z^{2}-12 z -1}{4 z^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.718 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.569 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {2 x^{2}-3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.82 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.545 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+2 y = 0 \] \(r = \frac {4 x^{2}-4 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.763 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+4 y = 0 \] \(r = x^{2}-3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_erf] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.516 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0 \] \(r = \frac {x \left (x^{3}+8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.638 |
|
|
\[ {}\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y = 0 \] \(r = \frac {-4 x^{2}+6}{\left (4 x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.913 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] \(r = \frac {15}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
0.802 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {1}{4} x^{2}-x -\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.542 |
|
|
\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \] \(r = \frac {5 x^{2}+6}{4 \left (2 x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.645 |
|
|
\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}}{64}-\frac {7}{8}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Lienard] |
✓ |
✓ |
0.57 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {9}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.165 |
|
|
\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x -32}{64 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.474 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \] \(r = \frac {324 x^{2}-252 x -35}{144 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.812 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \] \(r = \frac {x^{2}+16 x +40}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.006 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \] \(r = \frac {4 x^{2}-60 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.826 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (2 x +1\right ) y = 0 \] \(r = \frac {x^{2}-2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y = 0 \] \(r = \frac {4 x^{2}-4 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.703 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] \(r = \frac {x^{2}+10 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.772 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (-x +3\right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}-6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.684 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.867 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \] \(r = \frac {x^{2}-2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.64 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{2}+4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.662 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \] \(r = \frac {4 x^{2}-4 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.658 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] \(r = \frac {-x^{2}+8 x +8}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.773 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {7}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.747 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.632 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}+6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.671 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \] \(r = \frac {x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y = 0 \] \(r = \frac {x^{2}+12 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.747 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}-10 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.723 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.696 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] \(r = \frac {2+x}{x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \] \(r = \frac {x^{2}+2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.631 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.561 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.462 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.849 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.471 |
|
|
\[ {}2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y = 0 \] \(r = \frac {100 x^{2}-60 x +5}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.795 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0 \] \(r = \frac {n^{2}-2 n x +x^{2}-2 n -4 x}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.852 |
|
|
\[ {}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = \frac {-10 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0 \] \(r = \frac {4 x^{2}+4 x +15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.7 |
|
|
\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \] \(r = \frac {-12 x^{4}+156 x^{3}+297 x^{2}-78 x -3}{16 \left (2 x^{3}-7 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.622 |
|
|
\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.612 |
|
|
\[ {}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (-\frac {1}{4} x -x^{2}\right ) y^{\prime }-\frac {5 x y}{16} = 0 \] \(r = \frac {-192 x^{2}-36 x +9}{64 \left (4 x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.464 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] \(r = \frac {x^{2}-2 x +35}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.752 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] \(r = \frac {x^{2}-10 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.749 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}+8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.651 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = 0 \] \(r = \frac {5 x^{2}+36 x +4}{16 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0 \] \(r = \frac {-4 x^{2}+4 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.809 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {-3 x +8}{16 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.503 |
|
|
\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \] \(r = \frac {-3 x^{2}+66 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.845 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \] \(r = \frac {72 x^{2}-72 x -5}{36 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.86 |
|
|
\[ {}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0 \] \(r = \frac {3 x^{2}-6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.751 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.704 |
|
|
\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \] \(r = \frac {-a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.891 |
|
|
\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {a^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \] \(r = \frac {-n^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.55 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0 \] \(r = \frac {2 a^{2}-x^{2}}{x^{2} a^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.875 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.952 |
|
|
\[ {}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}} \] \(r = \frac {q^{2} x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.824 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.865 |
|
|
\[ {}x^{2} \left (2-x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {3}{\left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.781 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.58 |
|
|
\[ {}x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.621 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (-2+3 x \right ) y = 0 \] \(r = -\frac {2}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.908 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.806 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.793 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.68 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.389 |
|
|
\[ {}\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-8 x +18}{4 \left (2 x -3\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}}{4}+\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.575 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.531 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.514 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = \frac {4 x^{2}-4 x -3}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.877 |
|
|
\[ {}x \left (1+x \right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.697 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.805 |
|
|
\[ {}x y^{\prime \prime }+x y^{\prime }-2 y = 0 \] \(r = \frac {x +8}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.679 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+x^{2} y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0 \] \(r = \frac {x^{6}+2 x^{5}-5 x^{4}-16 x^{3}+24 x^{2}+24 x +12}{4 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.302 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-\left (2 x +1\right ) \left (-y+x y^{\prime }\right ) = 0 \] \(r = \frac {-4 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.884 |
|
|
\[ {}2 x^{2} \left (2-x \right ) y^{\prime \prime }-x \left (4-x \right ) y^{\prime }+\left (-x +3\right ) y = 0 \] \(r = -\frac {3}{16 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.755 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0 \] \(r = \frac {4-x}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.574 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.687 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.166 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.162 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \] \(r = \frac {12 x^{2}-13}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.93 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+2 \left (1+x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+4 x -1}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}x \left (2+x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}+30 x -3}{4 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.813 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.443 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}-32 x +180}{4 \left (x^{2}-2 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15 x^{2}-32 x +180}{4 \left (x^{2}-2 x +10\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.521 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.5 |
|
|
\[ {}\left (2+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}+4 x +12}{4 \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-6 y = 0 \] \(r = \frac {6}{x^{2}+1}\) \(L = [1, 4, 6, 12]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.431 |
|
|
\[ {}\left (x^{2}+2\right ) y^{\prime \prime }+3 x y^{\prime }-y = 0 \] \(r = \frac {7 x^{2}+20}{4 \left (x^{2}+2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.757 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \] \(r = x^{2}-9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.162 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3} = 0 \] \(r = \frac {9 x^{2}+30 x +7}{36 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.848 |
|
|
\[ {}2 x y^{\prime \prime }-y^{\prime }+2 y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.589 |
|
|
\[ {}2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y = 0 \] \(r = \frac {4 x^{2}+4 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.764 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (2 x -1\right ) y = 0 \] \(r = \frac {5-16 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.852 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.543 |
|
|
\[ {}x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}+48}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.776 |
|
|
\[ {}x^{4} y^{\prime \prime }+\lambda y = 0 \] \(r = -\frac {\lambda }{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.39 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y = 0 \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.546 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.363 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.444 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \] \(r = \frac {-4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.52 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.333 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \] \(r = \frac {-x^{4}+12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.551 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0 \] \(r = \frac {x \left (x^{3}-8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.408 |
|
|
\[ {}x y^{\prime \prime }-\left (2+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-4 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.389 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.313 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {2 x^{2}-3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.457 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.247 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y = 0 \] \(r = \frac {30 x^{2}-31}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
0.532 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.288 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {-3 x^{2}+18 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.75 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (-2+x \right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.596 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+\left (4 x^{2}+6\right ) y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.577 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.598 |
|
|
\[ {}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {48 x -3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[_Jacobi] |
✓ |
✓ |
0.915 |
|
|
\[ {}4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = \frac {-4 t^{2}+20 t -19}{16 \left (t^{2}-3 t +2\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.931 |
|
|
\[ {}2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y = 0 \] \(r = \frac {60 t^{2}-308 t +381}{16 \left (t^{2}-5 t +6\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}3 t \left (t +1\right ) y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {7 t +12}{36 t \left (t +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.883 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4} = 0 \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.609 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4} = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.585 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
0.638 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.847 |
|
|
\[ {}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0 \] \(r = -\frac {2}{x^{2} \left (x^{2}-1\right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.63 |
|
|
\[ {}2 x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}+4 x +12}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.665 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.483 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.526 |
|
|
\[ {}u^{\prime \prime }+2 u^{\prime }+u = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.231 |
|
|
\[ {}u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.494 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (1+3 x \right )^{2}}\right ) y = 0 \] \(r = -\frac {2}{\left (1+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.808 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.535 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}} = 0 \] \(r = \frac {2}{\left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.628 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.518 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.503 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.502 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.503 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.505 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }-x y = 0 \] \(r = \frac {1}{4} x^{2}+x -\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.504 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[_Lienard] |
✓ |
✓ |
0.483 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-x y = 0 \] \(r = \frac {8 x -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.7 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x^{2}+2 x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {9 x^{2}+12 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.688 |
|
|
\[ {}2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y = 0 \] \(r = \frac {21 x^{4}+18 x^{3}+27 x^{2}-2 x -3}{16 \left (x^{3}+x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.332 |
|
|
\[ {}x y^{\prime \prime }+\left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}-6 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.665 |
|
|
\[ {}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (x +4\right ) y = 0 \] \(r = \frac {7 x^{2}+10 x -1}{4 x^{2} \left (-1+x \right )^{4}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2, 4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y = 0 \] \(r = \frac {-3 x^{2}-24 x -16}{16 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.97 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.542 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.536 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y = 0 \] \(r = \frac {x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.74 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.519 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 y x^{4} = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.522 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.42 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.608 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.191 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{x^{2}} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime } = \frac {6 y}{x^{2}} \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.434 |
|
|
\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \] \(r = \frac {-32 x^{2}+27 x -27}{144 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.812 |
|
|
|
|||||
|
\[ {}y^{\prime \prime } = \frac {20 y}{x^{2}} \] \(r = \frac {20}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.573 |
|
|
\[ {}y^{\prime \prime } = \frac {12 y}{x^{2}} \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.592 |
|
|
\[ {}y^{\prime \prime }-\frac {y}{4 x^{2}} = 0 \] \(r = \frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.806 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.773 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{x^{2}} = 0 \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.884 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = \frac {4 x^{2}+4 x -3}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
78.523 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {4-x}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.416 |
|
|
\[ {}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0 \] \(r = \frac {14 x^{2}+5}{4 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.692 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}} \] \(r = \frac {4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.444 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {6}{x^{2}}-1\right ) y \] \(r = \frac {-x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.955 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y \] \(r = \frac {x^{2}}{4}-\frac {11}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.832 |
|
|
\[ {}y^{\prime \prime } = \left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y \] \(r = \frac {16 x -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (-1+x \right )^{2}}+\frac {3}{16 x \left (-1+x \right )}\right ) y \] \(r = \frac {-32 x^{2}+27 x -27}{144 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.623 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}} \] \(r = \frac {-5 x^{2}-27}{36 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
129.102 |
|
|
\[ {}y^{\prime \prime } = -\frac {y}{4 x^{2}} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.479 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}x^{2} y^{\prime \prime } = 2 y \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(6\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.584 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.444 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.563 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-\left (-2+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.688 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.749 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }+y-\sin \left (n x \right ) = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.756 |
|
|
\[ {}y^{\prime \prime }+y-a \cos \left (b x \right ) = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.768 |
|
|
\[ {}y^{\prime \prime }+y-\sin \left (x a \right ) \sin \left (b x \right ) = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.983 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.702 |
|
|
\[ {}y^{\prime \prime }-2 y-4 x^{2} {\mathrm e}^{x^{2}} = 0 \] \(r = 2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }+a^{2} y-\cot \left (x a \right ) = 0 \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.583 |
|
|
\[ {}y^{\prime \prime }+l y = 0 \] \(r = -l\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.927 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2}+1\right ) y = 0 \] \(r = x^{2}+1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.783 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0 \] \(r = \left (x^{2} a +1\right ) a\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.831 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y = 0 \] \(r = \frac {a^{2}}{4}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.394 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y-f \left (x \right ) = 0 \] \(r = \frac {a^{2}}{4}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.007 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = \frac {x^{2}}{4}-\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.614 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {x^{2}}{4}+\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.151 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = \frac {x^{2}}{4}-\frac {5}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[_Hermite] |
✓ |
✓ |
0.908 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = \frac {1}{4} x^{2}-x +\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y-{\mathrm e}^{x} = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.744 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.583 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y-{\mathrm e}^{x^{2}} = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.314 |
|
|
\[ {}y^{\prime \prime }+2 a x y^{\prime }+y a^{2} x^{2} = 0 \] \(r = a\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.2 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0 \] \(r = \frac {x \left (x^{3}-8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }-\left (1+x \right )^{2} y = 0 \] \(r = \frac {1}{4} x^{4}+x^{2}+x +1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.937 |
|
|
\[ {}y^{\prime \prime }-x^{2} \left (1+x \right ) y^{\prime }+x \left (x^{4}-2\right ) y = 0 \] \(r = \frac {x \left (x^{5}-2 x^{4}+x^{3}-6 x +4\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -6\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.093 |
|
|
\[ {}y^{\prime \prime }+x^{4} y^{\prime }-x^{3} y = 0 \] \(r = \frac {x^{3} \left (x^{5}+12\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -8\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.919 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } \sqrt {x}+\left (\frac {1}{4 \sqrt {x}}+\frac {x}{4}-9\right ) y-x \,{\mathrm e}^{-\frac {x^{\frac {3}{2}}}{3}} = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.799 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.681 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+\tan \left (x \right )+b y = 0 \] \(r = \frac {a^{2}}{4}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.553 |
|
|
\[ {}y^{\prime \prime }+2 a y^{\prime } \cot \left (x a \right )+\left (-a^{2}+b^{2}\right ) y = 0 \] \(r = -b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.75 |
|
|
\[ {}y^{\prime \prime }+f \left (x \right ) y^{\prime }+\left (\frac {f \left (x \right )^{2}}{4}+\frac {f^{\prime }\left (x \right )}{2}+a \right ) y = 0 \] \(r = -a\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.575 |
|
|
\[ {}x \left (y^{\prime \prime }+y\right )-\cos \left (x \right ) = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-y a \,x^{3} = 0 \] \(r = \frac {4 a \,x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.975 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y-{\mathrm e}^{x} = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.601 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+a x y = 0 \] \(r = -a\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}x y^{\prime \prime }-x y^{\prime }-y-x \left (1+x \right ) {\mathrm e}^{x} = 0 \] \(r = \frac {x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.437 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {x^{2}-2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
1.018 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }-2 \left (-1+x \right ) y = 0 \] \(r = \frac {9 x^{2}-6 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}x y^{\prime \prime }-2 \left (x a +b \right ) y^{\prime }+\left (x \,a^{2}+2 a b \right ) y = 0 \] \(r = \frac {b \left (b +1\right )}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.359 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = \frac {x^{3}-2 x^{2}-5 x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.277 |
|
|
\[ {}x y^{\prime \prime }-\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y = 0 \] \(r = \frac {x^{3}+2 x^{2}+7 x +4}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.05 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2} a +1\right ) y^{\prime }+b \,x^{3} y = 0 \] \(r = \frac {4 a^{2} x^{4}-4 x^{4} b +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.61 |
|
|
\[ {}x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5} = 0 \] \(r = \frac {32 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.376 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a \,x^{3}-1\right ) y^{\prime }+\left (a^{2} x^{3}+a \right ) x^{2} y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.217 |
|
|
\[ {}x y^{\prime \prime }+\left (2 a x \ln \left (x \right )+1\right ) y^{\prime }+\left (a^{2} x \ln \left (x \right )^{2}+a \ln \left (x \right )+a \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.072 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] \(r = \frac {4 x^{2}-12 x +15}{4 \left (x -3\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.245 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime }+a y = 0 \] \(r = \frac {-8 x a -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.266 |
|
|
\[ {}\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (x -3\right ) y = 0 \] \(r = \frac {x^{2}+4 x -6}{4 \left (2 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.496 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }-y = 0 \] \(r = \frac {-3+4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.116 |
|
|
\[ {}4 x y^{\prime \prime }+4 y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{2}+2 x -1}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.951 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.43 |
|
|
\[ {}x^{2} y^{\prime \prime }-12 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y = 0 \] \(r = -\frac {a}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {-x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.99 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2} a +2\right ) y = 0 \] \(r = \frac {x^{2} a +2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.076 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-6\right ) y = 0 \] \(r = \frac {-a^{2} x^{2}+6}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.29 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y = 0 \] \(r = \frac {4 b^{2} x^{4}+4 a b \,x^{2}+a^{2}-4 x a}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.299 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y-x^{2} a = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.91 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+a y = 0 \] \(r = \frac {-4 a -1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.013 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y-3 x^{3} = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.866 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-x^{5} \ln \left (x \right ) = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.727 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y-x \sin \left (x \right )-\left (x^{2} a +12 a +4\right ) \cos \left (x \right ) = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.401 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.854 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{2}}{\cos \left (x \right )} = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.411 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y-\frac {x^{3}}{\cos \left (x \right )} = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.013 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (a^{2} x^{2}+2\right ) y = 0 \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.944 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-2 x +1}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.895 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y-5 x = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.747 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }-5 y-\ln \left (x \right ) x^{2} = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.193 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y-x^{4}+x^{2} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.567 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y-\sin \left (x \right ) x^{3} = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.214 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0 \] \(r = \frac {a^{2}-2 a -4 b}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.714 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \] \(r = \frac {x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.79 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y = 0 \] \(r = \frac {x^{4}+2 x^{2}+4 x +1}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (x -9\right ) y = 0 \] \(r = \frac {x^{2}-2 x +35}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }+\left (3 x -1\right ) y = 0 \] \(r = \frac {x^{2}-10 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.962 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (-1+x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = \frac {x^{2}-6 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (3 x +2\right ) y = 0 \] \(r = \frac {x^{2}+8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.055 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (x +4\right ) y^{\prime }+4 y = 0 \] \(r = \frac {x^{2}+8 x +8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.819 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-4 y = 0 \] \(r = \frac {4 x^{2}+4 x +15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.879 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.089 |
|
|
\[ {}x^{2} y^{\prime \prime }+y^{\prime } x^{2} a -2 y = 0 \] \(r = \frac {a^{2} x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.892 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a +2 b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) b \,x^{2}-2\right ) y = 0 \] \(r = \frac {a^{2} x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.175 |
|
|
\[ {}x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {x^{4}-2 x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.312 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+\left (x^{2}-2\right ) y = 0 \] \(r = \frac {x^{4}+2 x^{2}+8}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.545 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y = 0 \] \(r = -\frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.19 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = \frac {-9 x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.028 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-9 y = 0 \] \(r = \frac {35 x^{2}+38}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.615 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] \(r = \frac {-4 x^{2} a -x^{2}-4 a +2}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.71 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.075 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.867 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+a y = 0 \] \(r = \frac {-4 x^{2} a +3 x^{2}-4 a +6}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.014 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y-2 \cos \left (x \right )+2 x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.898 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+2 = 0 \] \(r = \frac {-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.165 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] \(r = \frac {-4 x^{2} a -x^{2}+4 a -2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.128 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \] \(r = -\frac {1}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.871 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-a = 0 \] \(r = -\frac {1}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.484 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-\left (1+3 x \right ) y^{\prime }-\left (x^{2}-x \right ) y = 0 \] \(r = \frac {4 x^{4}-4 x^{3}+11 x^{2}+14 x +7}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.693 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.661 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 a x y^{\prime }+a \left (a -1\right ) y = 0 \] \(r = \frac {a \left (-2+a \right )}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Gegenbauer] |
✓ |
✓ |
1.091 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0 \] \(r = \frac {8 a^{2}}{\left (a^{2}-x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] \(r = \frac {-x^{2}-10 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.766 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }+\left (3 x +2\right ) y^{\prime }+y = 0 \] \(r = -\frac {1}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.415 |
|
|
\[ {}\left (x^{2}+x -2\right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }-\left (6 x^{2}+7 x \right ) y = 0 \] \(r = \frac {25 x^{3}+75 x^{2}+60 x -4}{\left (-4+4 x \right ) \left (2+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (x^{2}+x -1\right ) y^{\prime }-\left (2+x \right ) y = 0 \] \(r = \frac {x^{4}+6 x^{3}+17 x^{2}+26 x +15}{4 \left (1+x \right )^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.96 |
|
|
\[ {}x \left (x +3\right ) y^{\prime \prime }+\left (3 x -1\right ) y^{\prime }+y-\left (20 x +30\right ) \left (x^{2}+3 x \right )^{\frac {7}{3}} = 0 \] \(r = \frac {-x^{2}-14 x +7}{4 \left (x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.214 |
|
|
\[ {}\left (x^{2}+3 x +4\right ) y^{\prime \prime }+\left (x^{2}+x +1\right ) y^{\prime }-\left (2 x +3\right ) y = 0 \] \(r = \frac {x^{4}+10 x^{3}+43 x^{2}+82 x +51}{4 \left (x^{2}+3 x +4\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.81 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-\left (-2+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.362 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-\left (2 x^{2}+l -5 x \right ) y^{\prime }-\left (4 x -1\right ) y = 0 \] \(r = \frac {4 x^{4}+4 l \,x^{2}+12 x^{3}+l^{2}-2 l x -3 x^{2}}{16 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 0\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.983 |
|
|
\[ {}\left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y = 0 \] \(r = \frac {4 x^{4}-4 x^{3}-27 x^{2}-32 x +8}{16 \left (x^{2}+3 x +2\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.766 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.299 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (4 x^{2}+1\right ) y-4 \sqrt {x^{3}}\, {\mathrm e}^{x} = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.011 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-\left (x^{2} a +1\right ) y = 0 \] \(r = \frac {a}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.624 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+5 x y^{\prime }-y-\ln \left (x \right ) = 0 \] \(r = \frac {1}{64 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.365 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }-\left (4 x^{2}+12 x +3\right ) y = 0 \] \(r = \frac {4 x^{2}+12 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.931 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (2 x -1\right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.582 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+6\right ) \left (x^{2}-4\right ) y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (x^{2} \ln \left (x \right )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}} = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.788 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y-3 x -1 = 0 \] \(r = \frac {15}{\left (2 x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (3 x -1\right )^{2} y^{\prime \prime }+3 \left (3 x -1\right ) y^{\prime }-9 y-\ln \left (3 x -1\right )^{2} = 0 \] \(r = \frac {27}{4 \left (3 x -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.682 |
|
|
\[ {}9 x \left (-1+x \right ) y^{\prime \prime }+3 \left (2 x -1\right ) y^{\prime }-20 y = 0 \] \(r = \frac {72 x^{2}-72 x -5}{36 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
0.776 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+\left (3+4 x \right ) y = 0 \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.473 |
|
|
\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }-\left (5+4 x \right ) y = 0 \] \(r = \frac {5+4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.667 |
|
|
\[ {}\left (27 x^{2}+4\right ) y^{\prime \prime }+27 x y^{\prime }-3 y = 0 \] \(r = \frac {-405 x^{2}+264}{4 \left (27 x^{2}+4\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.43 |
|
|
\[ {}50 x \left (-1+x \right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y = 0 \] \(r = \frac {-84 x^{2}+84 x -75}{400 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.267 |
|
|
\[ {}\left (x^{2} a +1\right ) y^{\prime \prime }+a x y^{\prime }+b y = 0 \] \(r = \frac {-a^{2} x^{2}-4 a b \,x^{2}+2 a -4 b}{4 \left (x^{2} a +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(4\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
187.713 |
|
|
\[ {}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime } = 0 \] \(r = -\frac {a^{2}}{\left (a^{2} x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.988 |
|
|
\[ {}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y = 0 \] \(r = \frac {a^{2} \left (2 a^{2} x^{2}-3\right )}{\left (a^{2} x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
1.786 |
|
|
\[ {}\left (x^{2} a +b x \right ) y^{\prime \prime }+2 b y^{\prime }-2 a y = 0 \] \(r = \frac {2 a^{2}}{\left (x a +b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.225 |
|
|
\[ {}x^{3} y^{\prime \prime }+x y^{\prime }-\left (2 x +3\right ) y = 0 \] \(r = \frac {8 x^{2}+8 x +1}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.037 |
|
|
\[ {}x^{3} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-2 y = 0 \] \(r = \frac {-x^{2}+6 x +1}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.124 |
|
|
\[ {}x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y-\ln \left (x \right )^{3} = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.979 |
|
|
\[ {}x^{3} y^{\prime \prime }+3 x^{2} y^{\prime }+x y-1 = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.872 |
|
|
\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+2 \left (x^{2}-1\right ) y^{\prime }-2 x y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.48 |
|
|
\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime }+y a \,x^{3} = 0 \] \(r = \frac {-4 a \,x^{6}+4 a \,x^{4}-6 x^{2}+3}{4 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.132 |
|
|
\[ {}x \left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime }-6 x y = 0 \] \(r = \frac {24 x^{4}+54 x^{2}+5}{4 \left (x^{3}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.017 |
|
|
\[ {}x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y = 0 \] \(r = \frac {x^{6}+2 x^{5}-5 x^{4}-16 x^{3}+24 x^{2}+24 x +12}{4 \left (x^{3}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.775 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (2 x +1\right ) y = 0 \] \(r = \frac {-4 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.434 |
|
|
\[ {}x^{2} \left (1+x \right ) y^{\prime \prime }+2 x \left (3 x +2\right ) y^{\prime } = 0 \] \(r = \frac {6 x +2}{x^{2} \left (1+x \right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (-2+x \right ) y^{\prime }}{x \left (-1+x \right )}+\frac {2 \left (1+x \right ) y}{x^{2} \left (-1+x \right )} \] \(r = \frac {2}{\left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.983 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (-1+x \right )} \] \(r = \frac {4-x}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.243 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{1+x}-\frac {y}{x \left (1+x \right )^{2}} \] \(r = \frac {-x -4}{4 x \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.036 |
|
|
\[ {}y^{\prime \prime } = \frac {2 y}{x \left (-1+x \right )^{2}} \] \(r = \frac {2}{x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (-2+x \right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (-2+x \right )} \] \(r = -\frac {3}{16 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.705 |
|
|
\[ {}y^{\prime \prime } = \frac {y^{\prime }}{1+x}-\frac {\left (1+3 x \right ) y}{4 x^{2} \left (1+x \right )} \] \(r = \frac {-4 x -1}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.703 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (1-3 x \right ) y}{\left (-1+x \right ) \left (2 x -1\right )^{2}} \] \(r = \frac {3 x -1}{\left (-1+x \right ) \left (2 x -1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
8.41 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}-\frac {\left (-b +a \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )} \] \(r = -\frac {3}{16 \left (x +b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.638 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (-2+x \right )}+\frac {y}{3 x^{2} \left (-2+x \right )} \] \(r = \frac {72 x^{2}-12 x -11}{36 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.767 |
|
|
\[ {}y^{\prime \prime } = \frac {2 \left (x a +2 b \right ) y^{\prime }}{x \left (x a +b \right )}-\frac {\left (2 x a +6 b \right ) y}{\left (x a +b \right ) x^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.803 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{x^{4}} \] \(r = -\frac {a}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}} \] \(r = \frac {2 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.438 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (x \left (a +b \right )+a b \right ) y}{x^{4}} \] \(r = \frac {a^{2}-2 a b +b^{2}}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.729 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}} \] \(r = -\frac {a^{2}}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.519 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}} \] \(r = \frac {2 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.283 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}} \] \(r = \frac {a^{2}-b}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.389 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \] \(r = \frac {8 x^{4}-14 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.269 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}} \] \(r = \frac {8 x^{4}-18 x^{2}+1}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.865 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1} \] \(r = \frac {3 x^{6}+14 x^{3}+3}{4 \left (x^{4}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.352 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )} \] \(r = \frac {-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (1+a \right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \] \(r = \frac {a^{2} x^{4}+3 a \,x^{4}-2 a^{2} x^{2}+2 x^{4}-4 x^{2} a +a^{2}+x^{2}+a}{\left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
54.459 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}} \] \(r = -\frac {a}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Halm] |
✓ |
✓ |
0.843 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.445 |
|
|
\[ {}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}} \] \(r = -\frac {a}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}} \] \(r = \frac {a^{2}-1}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.851 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )} \] \(r = \frac {2 x^{2} a -4 b \,x^{2}-a^{2}-4 a b}{4 \left (x^{3}+x a \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.229 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}} \] \(r = -\frac {b^{2}}{\left (a^{2}+x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.93 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (-1+x \right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (-1+x \right )^{2}} \] \(r = \frac {2}{x^{2}-x}\) \(L = [1, 4, 6, 12]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime } = \frac {12 y}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )} \] \(r = \frac {12}{\left (1+x \right )^{2} \left (x^{2}+2 x +3\right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.257 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}} \] \(r = -\frac {b}{\left (x a -x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.246 |
|
|
\[ {}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c \] \(r = -\frac {b}{\left (x a -x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
54.631 |
|
|
\[ {}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] \(r = \frac {c}{\left (a b -x a -b x +x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.08 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (-b +x \right )+\left (1-\alpha -\beta \right ) \left (-b +x \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (-b +x \right )^{2}}-\frac {\alpha \beta \left (-b +a \right )^{2} y}{\left (x -a \right )^{2} \left (-b +x \right )^{2}} \] \(r = \frac {a^{2} \alpha ^{2}-2 \alpha \,a^{2} \beta +a^{2} \beta ^{2}-2 a \,\alpha ^{2} b +4 b \alpha a \beta -2 a b \,\beta ^{2}+\alpha ^{2} b^{2}-2 \alpha \,b^{2} \beta +b^{2} \beta ^{2}-a^{2}+2 a b -b^{2}}{4 \left (a b -x a -b x +x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.648 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a +a -3\right ) y}{4 \left (x^{2}+1\right )^{2}} \] \(r = \frac {-x^{2} a -a +3}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Halm] |
✓ |
✓ |
3.441 |
|
|
\[ {}y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )} \] \(r = \frac {18}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.484 |
|
|
\[ {}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}} \] \(r = \frac {3}{4 \left (x^{2}+x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.257 |
|
|
\[ {}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (-1+x \right )^{2}} \] \(r = -\frac {3}{16 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (x a +b \right )^{2}} \] \(r = -\frac {c}{\left (x^{2} a +b x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.206 |
|
|
\[ {}y^{\prime \prime } = -\frac {y}{\left (x a +b \right )^{4}} \] \(r = -\frac {1}{\left (x a +b \right )^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.93 |
|
|
|
|||||
|
\[ {}y^{\prime \prime } = -\frac {A y}{\left (x^{2} a +b x +c \right )^{2}} \] \(r = -\frac {A}{\left (x^{2} a +b x +c \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.532 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}} \] \(r = \frac {-4 x^{3}+1}{4 x^{8}}\) \(L = [1]\) case used \(1\) poles order = \([8]\) \( O(\infty ) = 5\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.842 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (1+3 x \right ) y^{\prime }}{\left (-1+x \right ) \left (1+x \right )}-\frac {36 \left (1+x \right )^{2} y}{\left (-1+x \right )^{2} \left (3 x +5\right )^{2}} \] \(r = \frac {-9 x^{4}-36 x^{3}-126 x^{2}-116 x +31}{4 \left (3 x^{3}+5 x^{2}-3 x -5\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.965 |
|
|
\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.464 |
|
|
\[ {}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}} \] \(r = \frac {8-a}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.635 |
|
|
\[ {}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}} \] \(r = -\frac {27 x}{16 \left (x^{3}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 5\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
80.596 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-1+x \right ) y}{x^{4}} \] \(r = \frac {-x^{2}-4 x +4}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}} \] \(r = \frac {-x^{2}+4 x +4}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime } = -\frac {b^{2} y}{\left (-a^{2}+x^{2}\right )^{2}} \] \(r = -\frac {b^{2}}{\left (a^{2}-x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.463 |
|
|
\[ {}y^{\prime \prime }+a y = 0 \] \(r = -a\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.442 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0 \] \(r = a \left (x^{2} a +1\right )\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }+a^{3} x \left (-x a +2\right ) y = 0 \] \(r = x \,a^{3} \left (x a -2\right )\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.855 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y = 0 \] \(r = \frac {a^{2}}{4}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+x a +1\right ) y = 0 \] \(r = b^{2} x^{2}-a b x +\frac {1}{4} a^{2}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b x \left (-b \,x^{3}+x a +2\right ) y = 0 \] \(r = b^{2} x^{4}-a b \,x^{2}+\frac {1}{4} a^{2}-2 b x\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.838 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }-a y = 0 \] \(r = \frac {3}{2} a +\frac {1}{4} a^{2} x^{2}+\frac {1}{2} a b x +\frac {1}{4} b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.144 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+a y = 0 \] \(r = -\frac {1}{2} a +\frac {1}{4} a^{2} x^{2}+\frac {1}{2} a b x +\frac {1}{4} b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.445 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (x a +b -c \right ) y = 0 \] \(r = \frac {1}{2} a +\frac {1}{4} a^{2} x^{2}+\frac {1}{2} a b x +\frac {1}{4} b^{2}-a c x -b c +c^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}y^{\prime \prime }+\left (x a +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y = 0 \] \(r = \frac {a \left (x^{2} a +6\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime }+2 \left (x a +b \right ) y^{\prime }+\left (a^{2} x^{2}+2 a b x +c \right ) y = 0 \] \(r = b^{2}+a -c\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.745 |
|
|
\[ {}y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y = 0 \] \(r = \frac {a \left (a \,b^{4}-2 a \,b^{2} x^{2}+a \,x^{4}+4 b +8 x \right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.428 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c \left (x^{2} a +b -c \right ) y = 0 \] \(r = x a +\frac {1}{4} a^{2} x^{4}+\frac {1}{2} a b \,x^{2}+\frac {1}{4} b^{2}-a c \,x^{2}-b c +c^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.267 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +2 b \right ) y^{\prime }+\left (a b \,x^{2}-x a +b^{2}\right ) y = 0 \] \(r = \frac {x a \left (a \,x^{3}+8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+x^{2} a +b +2 x \right ) y = 0 \] \(r = \frac {a^{2}}{4}-b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.814 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y = 0 \] \(r = \frac {b \left (a^{2} b \,x^{4}+2 a b \,x^{3}+b \,x^{2}+8 x a +2\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.575 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y = 0 \] \(r = -x a +\frac {1}{2} b +\frac {1}{4} a^{2} x^{4}-\frac {1}{2} a b \,x^{3}+\frac {1}{2} a c \,x^{2}+\frac {1}{4} b^{2} x^{2}-\frac {1}{2} c b x +\frac {1}{4} c^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.523 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y = 0 \] \(r = x a -\frac {1}{2} b +\frac {1}{4} a^{2} x^{4}-\frac {1}{2} a b \,x^{3}-\frac {1}{2} a c \,x^{2}+\frac {1}{4} b^{2} x^{2}+\frac {1}{2} c b x +\frac {1}{4} c^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-x^{2} a +b^{2}\right ) y = 0 \] \(r = \frac {x^{2} a \left (a \,x^{4}+10\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -6\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 x^{2} a +b \right ) y = 0 \] \(r = -\frac {5}{2} x^{2} a -\frac {3}{2} b +\frac {1}{4} a^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{4} b^{2} x^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -6\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.715 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{3}+1\right ) y = 0 \] \(r = \frac {b x \left (a^{2} b \,x^{5}+2 a b \,x^{4}+b \,x^{3}+10 x a +4\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -6\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.619 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+a y = 0 \] \(r = \frac {-16 x a -3}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.139 |
|
|
\[ {}x y^{\prime \prime }+a x y^{\prime }+a y = 0 \] \(r = \frac {a \left (x a -4\right )}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 0\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.615 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a \left (x a +b \right ) y = 0 \] \(r = \frac {b \left (b -2\right )}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.948 |
|
|
\[ {}x y^{\prime \prime }-\left (x a +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y = 0 \] \(r = \frac {4 b^{2} x^{4}+4 a b \,x^{3}+a^{2} x^{2}+2 x a +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.168 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x a +1\right ) y^{\prime }+\left (b \,x^{3}+x \,a^{2}+a \right ) y = 0 \] \(r = \frac {-4 b \,x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.494 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }-\left (a c \,x^{2}+\left (b c +c^{2}+a \right ) x +b +2 c \right ) y = 0 \] \(r = \frac {a^{2} x^{3}+2 a b \,x^{2}+4 a c \,x^{2}+b^{2} x +4 c b x +4 c^{2} x +6 x a +4 b +8 c}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.257 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +2\right ) y^{\prime }+b y = 0 \] \(r = \frac {3}{2} a +\frac {1}{4} a^{2} x^{2}+\frac {1}{2} a b x +\frac {1}{4} b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.163 |
|
|
\[ {}x y^{\prime \prime }+x \left (x^{2} a +b \right ) y^{\prime }+\left (3 x^{2} a +b \right ) y = 0 \] \(r = \frac {a^{2} x^{5}+2 a b \,x^{3}-8 x^{2} a +b^{2} x -4 b}{4 x}\) \(L = [1]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = -4\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+y b x = 0 \] \(r = \frac {1}{4} a^{2} x^{4}+\frac {1}{2} a b \,x^{3}+\frac {1}{4} b^{2} x^{2}+2 x a +\frac {1}{2} b\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.859 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+x a -1\right ) y^{\prime }+a^{2} y b \,x^{3} = 0 \] \(r = \frac {a^{2} b^{2} x^{6}+2 a \,b^{2} x^{5}-2 a^{2} b \,x^{4}+b^{2} x^{4}+4 a b \,x^{3}+a^{2} x^{2}-2 x a +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.772 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y = 0 \] \(r = -\frac {a}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.561 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y = 0 \] \(r = \frac {16 a \,x^{3}+5}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = -1\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.743 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0 \] \(r = \frac {a^{2}-2 a -4 b}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.983 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-\left (a^{2} x^{2}+2\right ) y = 0 \] \(r = \frac {a^{2} x^{2}+2}{x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.952 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (1+a \right )\right ) y = 0 \] \(r = -b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.911 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (1+a \right )\right ) y = 0 \] \(r = b^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.858 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0 \] \(r = \frac {a^{2} b^{2} x^{2}-2 b \,a^{2} x^{3}+a^{2} x^{4}+2 a \,b^{2} x -2 a b \,x^{2}-2 a \,x^{3}+b^{2}-6 b x +3 x^{2}}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.062 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] \(r = \frac {-4 x^{2} a -x^{2}+4 a -2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.435 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+n^{2} y = 0 \] \(r = \frac {4 n^{2} x^{2}-4 n^{2}-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.413 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (n +2\right ) y = 0 \] \(r = \frac {4 n^{2} x^{2}+8 n \,x^{2}-4 n^{2}+3 x^{2}-8 n -6}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer] |
✓ |
✓ |
1.242 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0 \] \(r = \frac {-a^{2} x^{2}-4 a c \,x^{2}+2 a b -4 b c}{4 \left (x^{2} a +b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(4\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
146.921 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 y^{\prime } b x +b \left (b -1\right ) y = 0 \] \(r = -\frac {a^{2} b \left (b -2\right )}{\left (a^{2}+x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (2 x a +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0 \] \(r = \frac {8 a \,m^{2} x +4 m^{2} x^{2}+4 b \,m^{2}-3 a^{2}-2 x a -x^{2}+2 b}{4 \left (2 x a +x^{2}+b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.022 |
|
|
\[ {}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+d y = 0 \] \(r = \frac {-a^{2} x^{2}-4 a d \,x^{2}-2 a b x -8 b d x +2 a c -3 b^{2}-4 c d}{4 \left (x^{2} a +2 b x +c \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
4.695 |
|
|
\[ {}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+3 \left (x a +b \right ) y^{\prime }+d y = 0 \] \(r = \frac {3 a^{2} x^{2}-4 a d \,x^{2}+6 a b x -8 b d x +6 a c -3 b^{2}-4 c d}{4 \left (x^{2} a +2 b x +c \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.126 |
|
|
\[ {}x \left (x^{2} a +b \right ) y^{\prime \prime }+2 \left (x^{2} a +b \right ) y^{\prime }-2 a x y = 0 \] \(r = \frac {2 a}{x^{2} a +b}\) \(L = [1, 4, 6, 12]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.434 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }-2 x \left (x a +2 b \right ) y^{\prime }+2 \left (x a +3 b \right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.109 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (1+m \right )\right ) y = 0 \] \(r = \frac {m^{2}-2 m n +n^{2}+2 m -2 n}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.351 |
|
|
\[ {}2 x \left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (x^{2} a -c \right ) y^{\prime }+\lambda \,x^{2} y = 0 \] \(r = \frac {-8 a \lambda \,x^{5}-3 a^{2} x^{4}-8 b \lambda \,x^{4}-8 c \lambda \,x^{3}+14 a c \,x^{2}+8 c b x +5 c^{2}}{16 \left (a \,x^{3}+b \,x^{2}+c x \right )^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
190.64 |
|
|
\[ {}x^{4} y^{\prime \prime }+a y = 0 \] \(r = -\frac {a}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.735 |
|
|
\[ {}x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (x \left (a +b \right )+a b \right ) y = 0 \] \(r = \frac {a^{2}-2 a b +b^{2}}{4 x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.055 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0 \] \(r = \frac {a^{2}-b}{x^{4}}\) \(L = [1]\) case used \(1\) poles order = \([4]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.009 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = 0 \] \(r = -\frac {b}{\left (x a -x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.817 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = c \,x^{2} \left (x -a \right )^{2} \] \(r = -\frac {b}{\left (x a -x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
24.363 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0 \] \(r = -\frac {a}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[_Halm] |
✓ |
✓ |
1.322 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0 \] \(r = -\frac {a}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.63 |
|
|
\[ {}\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+y b^{2} = 0 \] \(r = -\frac {b^{2}}{\left (a^{2}+x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.669 |
|
|
\[ {}\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+y b^{2} = 0 \] \(r = -\frac {b^{2}}{\left (a^{2}-x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.929 |
|
|
\[ {}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (x^{2} a +a -3\right ) y = 0 \] \(r = \frac {-x^{2} a -a +3}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Halm] |
✓ |
✓ |
3.16 |
|
|
\[ {}\left (x^{2} a +b \right )^{2} y^{\prime \prime }+2 a x \left (x^{2} a +b \right ) y^{\prime }+c y = 0 \] \(r = \frac {a b -c}{\left (x^{2} a +b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.989 |
|
|
\[ {}\left (x^{2} a +b \right )^{2} y^{\prime \prime }+\left (2 x a +c \right ) \left (x^{2} a +b \right ) y^{\prime }+k y = 0 \] \(r = \frac {4 a b +c^{2}-4 k}{4 \left (x^{2} a +b \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.594 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }-c y = 0 \] \(r = \frac {c}{\left (a b -x a -b x +x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.921 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (-b +x \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y = 0 \] \(r = \frac {4 a b +2 a \lambda +2 b \lambda +\lambda ^{2}-4 \mu }{4 \left (a b -x a -b x +x^{2}\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.727 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} y^{\prime \prime }+y A = 0 \] \(r = -\frac {A}{\left (x^{2} a +b x +c \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.69 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} y^{\prime \prime }+\left (2 x a +k \right ) \left (x^{2} a +b x +c \right ) y^{\prime }+m y = 0 \] \(r = \frac {4 a c -2 b k +k^{2}-4 m}{4 \left (x^{2} a +b x +c \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.501 |
|
|
\[ {}y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\lambda x} a +\lambda \right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.229 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{{\mathrm e}^{x}} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{\left (1-x \right )^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.519 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.42 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.739 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+4 y = x^{2}+\cos \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.291 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \,{\mathrm e}^{2 x} x -\sin \left (x \right )^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.616 |
|
|
\[ {}y^{\prime \prime }+y = 2 \,{\mathrm e}^{x}+x^{3}-x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.648 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 3 \,{\mathrm e}^{2 x}-\cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = {\mathrm e}^{2 x}+1 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.929 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{\left (1-x \right )^{2}} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.465 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-\left (1+x \right ) y^{\prime }+6 y = x \] \(r = -\frac {21}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.039 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \cos \left (x \right )-{\mathrm e}^{2 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 x^{3}-x \,{\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.601 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.099 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.996 |
|
|
\[ {}y^{\prime \prime }+y = x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.797 |
|
|
\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = x \] \(r = \frac {x \left (x^{3}-8\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.58 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = x^{2}-x -1 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2 x^{2}+3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.725 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2} \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.987 |
|
|
\[ {}\sin \left (x \right ) y^{\prime \prime }+2 \cos \left (x \right ) y^{\prime }+3 \sin \left (x \right ) y = {\mathrm e}^{x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.663 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0 \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.949 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }-x y = 2 \,{\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.154 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y = 0 \] \(r = \frac {15 x^{2}-18}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.29 |
|
|
\[ {}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = \frac {1}{x^{2}} \] \(r = \frac {3 x^{4}-4}{4 x^{6}}\) \(L = [1]\) case used \(1\) poles order = \([6]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.721 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime }-8 x^{3} y = 4 x^{3} {\mathrm e}^{-x^{2}} \] \(r = \frac {36 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.59 |
|
|
\[ {}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0 \] \(r = \frac {x^{2}-6 x +15}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[_Laguerre] |
✓ |
✓ |
2.465 |
|
|
\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+\left (3 x -6\right ) y = 0 \] \(r = \frac {4 x^{2}-12 x +15}{4 \left (x -3\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.355 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.87 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.629 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.741 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.934 |
|
|
\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0 \] \(r = \frac {3 x \left (x^{3}+4\right )}{\left (2 x^{3}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.394 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 \left (1+x \right ) y = x^{3} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.515 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime } = x \] \(r = \frac {x^{2}}{4}+\frac {1}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.628 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.692 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = x \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.149 |
|
|
\[ {}\left (-1+x \right )^{2} y^{\prime \prime }+4 \left (-1+x \right ) y^{\prime }+2 y = \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.064 |
|
|
\[ {}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0 \] \(r = \frac {8 x^{6}+1}{4 x^{8}}\) \(L = [1]\) case used \(1\) poles order = \([8]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.261 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2 \] \(r = \frac {-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.341 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (4 x +2\right ) y^{\prime }+2 y = 0 \] \(r = \frac {4 x +2}{\left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.688 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.316 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-\frac {y^{\prime }}{x}+x^{2} = 0 \] \(r = \frac {-6 x^{2}+3}{4 \left (x^{3}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.49 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+2 x = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.426 |
|
|
\[ {}t^{2} x^{\prime \prime }-6 x = 0 \] \(r = \frac {6}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.434 |
|
|
\[ {}2 x^{\prime \prime }-5 x^{\prime }-3 x = 0 \] \(r = {\frac {49}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.263 |
|
|
\[ {}x^{\prime \prime } = -3 \sqrt {t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.354 |
|
|
\[ {}x^{\prime }+t x^{\prime \prime } = 1 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.283 |
|
|
\[ {}\frac {x^{\prime }+t x^{\prime \prime }}{t} = -2 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.619 |
|
|
\[ {}x^{\prime \prime }+x^{\prime } = 3 t \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.671 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime }+4 x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.587 |
|
|
\[ {}x^{\prime \prime }-2 x^{\prime } = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.146 |
|
|
\[ {}\frac {x^{\prime \prime }}{2}+x^{\prime }+\frac {x}{2} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.597 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+3 x = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.479 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime }+4 x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.548 |
|
|
\[ {}x^{\prime \prime }-2 x^{\prime } = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.262 |
|
|
\[ {}\frac {x^{\prime \prime }}{2}+x^{\prime }+\frac {x}{2} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.575 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+3 x = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.453 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+4 x = 0 \] \(r = -{\frac {15}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.185 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime }+6 x = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.036 |
|
|
\[ {}x^{\prime \prime }+9 x = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.809 |
|
|
\[ {}x^{\prime \prime }-12 x = 0 \] \(r = 12\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.975 |
|
|
\[ {}2 x^{\prime \prime }+3 x^{\prime }+3 x = 0 \] \(r = -{\frac {15}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.132 |
|
|
\[ {}\frac {x^{\prime \prime }}{2}+\frac {5 x^{\prime }}{6}+\frac {2 x}{9} = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.487 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.422 |
|
|
\[ {}x^{\prime \prime }+\frac {x^{\prime }}{8}+x = 0 \] \(r = -{\frac {255}{256}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.143 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 3 t^{3}-1 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.739 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 3 \cos \left (t \right )-2 \sin \left (t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.902 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 12 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.627 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = t^{2} {\mathrm e}^{3 t} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.733 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 5 \sin \left (7 t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.403 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = {\mathrm e}^{2 t} \cos \left (t \right )+t^{2} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.605 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = t \,{\mathrm e}^{-t} \sin \left (\pi t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.656 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = \left (2+t \right ) \sin \left (\pi t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.636 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 4 t +5 \,{\mathrm e}^{-t} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.781 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 5 \sin \left (2 t \right )+t \,{\mathrm e}^{t} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.425 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = t^{3}+1-4 t \cos \left (t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.012 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = -6+2 \,{\mathrm e}^{2 t} \sin \left (t \right ) \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.516 |
|
|
\[ {}x^{\prime \prime }+7 x = t \,{\mathrm e}^{3 t} \] \(r = -7\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.832 |
|
|
\[ {}x^{\prime \prime }-x^{\prime } = 6+{\mathrm e}^{2 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.875 |
|
|
\[ {}x^{\prime \prime }+x = t^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.579 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }-4 x = 2 t^{2} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.452 |
|
|
\[ {}x^{\prime \prime }+x = 9 \,{\mathrm e}^{-t} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.61 |
|
|
\[ {}x^{\prime \prime }-4 x = \cos \left (2 t \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.682 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+2 x = t \sin \left (2 t \right ) \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.833 |
|
|
\[ {}x^{\prime \prime }-b x^{\prime }+x = \sin \left (2 t \right ) \] \(r = \frac {b^{2}}{4}-1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.822 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }-40 x = 2 \,{\mathrm e}^{-t} \] \(r = {\frac {169}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.705 |
|
|
\[ {}x^{\prime \prime }-2 x^{\prime } = 4 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.99 |
|
|
\[ {}x^{\prime \prime }+2 x = \cos \left (\sqrt {2}\, t \right ) \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.481 |
|
|
\[ {}x^{\prime \prime }+\frac {x^{\prime }}{100}+4 x = \cos \left (2 t \right ) \] \(r = -{\frac {159999}{40000}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.072 |
|
|
\[ {}x^{\prime \prime }+w^{2} x = \cos \left (\beta t \right ) \] \(r = -w^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.326 |
|
|
\[ {}x^{\prime \prime }+3025 x = \cos \left (45 t \right ) \] \(r = -3025\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.223 |
|
|
\[ {}x^{\prime \prime } = -\frac {x}{t^{2}} \] \(r = -\frac {1}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.654 |
|
|
\[ {}x^{\prime \prime } = \frac {4 x}{t^{2}} \] \(r = \frac {4}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.582 |
|
|
\[ {}t^{2} x^{\prime \prime }+3 t x^{\prime }+x = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.817 |
|
|
\[ {}t x^{\prime \prime }+4 x^{\prime }+\frac {2 x}{t} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.08 |
|
|
\[ {}t^{2} x^{\prime \prime }-7 t x^{\prime }+16 x = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.664 |
|
|
\[ {}t^{2} x^{\prime \prime }+3 t x^{\prime }-8 x = 0 \] \(r = \frac {35}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.609 |
|
|
\[ {}t^{2} x^{\prime \prime }+t x^{\prime } = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.187 |
|
|
\[ {}t^{2} x^{\prime \prime }-t x^{\prime }+2 x = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.863 |
|
|
\[ {}x^{\prime \prime }+t^{2} x^{\prime } = 0 \] \(r = \frac {t \left (t^{3}+4\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.904 |
|
|
\[ {}x^{\prime \prime }+x = \tan \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.783 |
|
|
\[ {}x^{\prime \prime }-x = t \,{\mathrm e}^{t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.494 |
|
|
\[ {}x^{\prime \prime }-x = \frac {1}{t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.47 |
|
|
\[ {}t^{2} x^{\prime \prime }-2 x = t^{3} \] \(r = \frac {2}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.755 |
|
|
\[ {}x^{\prime \prime }+x = \frac {1}{t +1} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}x^{\prime \prime }-2 x^{\prime }+x = \frac {{\mathrm e}^{t}}{2 t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.562 |
|
|
\[ {}\frac {x^{\prime }}{t}+x^{\prime \prime } = a \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.708 |
|
|
\[ {}t^{2} x^{\prime \prime }-3 t x^{\prime }+3 x = 4 t^{7} \] \(r = \frac {3}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.57 |
|
|
\[ {}x^{\prime \prime }-x = \frac {{\mathrm e}^{t}}{1+{\mathrm e}^{t}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.261 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 4 x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.426 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.784 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.263 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = -8 \sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.732 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
2.651 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
1.171 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.199 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.18 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.265 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.569 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.202 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.202 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+4 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.271 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 4 x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.42 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2-12 x +6 \,{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.279 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+5 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.282 |
|
|
\[ {}3 y^{\prime \prime }-14 y^{\prime }-5 y = 0 \] \(r = {\frac {64}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.31 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.336 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.342 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.665 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+25 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime }+9 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.802 |
|
|
\[ {}4 y^{\prime \prime }+y = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.345 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.5 |
|
|
\[ {}3 y^{\prime \prime }+4 y^{\prime }-4 y = 0 \] \(r = {\frac {16}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.689 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.688 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.716 |
|
|
\[ {}9 y^{\prime \prime }-6 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.661 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+29 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+58 y = 0 \] \(r = -49\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.927 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.89 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.859 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+5 y = 0 \] \(r = -{\frac {4}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.894 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+37 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.871 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+8 y = 4 x^{2} \] \(r = -{\frac {23}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.038 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 4 \,{\mathrm e}^{2 x}-21 \,{\mathrm e}^{-3 x} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 6 \sin \left (2 x \right )+7 \cos \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.03 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 10 \sin \left (4 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.043 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+4 y = \cos \left (4 x \right ) \] \(r = -3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.674 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-4 y = 16 x -12 \,{\mathrm e}^{2 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+5 y = 2 \,{\mathrm e}^{x}+10 \,{\mathrm e}^{5 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 5 \,{\mathrm e}^{-2 x} x \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.924 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 10 \,{\mathrm e}^{2 x}-18 \,{\mathrm e}^{3 x}-6 x -11 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 6 \,{\mathrm e}^{-2 x}+3 \,{\mathrm e}^{x}-4 x^{2} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }+y = x \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.943 |
|
|
\[ {}y^{\prime \prime }+4 y = 12 x^{2}-16 x \cos \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.52 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 9 x^{2}+4 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = 16 x +20 \,{\mathrm e}^{x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.769 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 9 \,{\mathrm e}^{2 x} x \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = 4 x \,{\mathrm e}^{-3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.791 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+16 y = 8 \,{\mathrm e}^{-2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.896 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 27 \,{\mathrm e}^{-6 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = 18 \,{\mathrm e}^{-2 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.211 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+29 y = 8 \,{\mathrm e}^{5 x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.202 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 8 \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.142 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 8 \,{\mathrm e}^{2 x}-5 \,{\mathrm e}^{3 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.992 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2 \,{\mathrm e}^{2 x} x +6 \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.05 |
|
|
\[ {}y^{\prime \prime }-y = 3 x^{2} {\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime }+y = 3 x^{2}-4 \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.307 |
|
|
\[ {}y^{\prime \prime }+4 y = 8 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.184 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = x^{3}+x +{\mathrm e}^{-2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.533 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{3 x}+{\mathrm e}^{-3 x}+{\mathrm e}^{3 x} \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.56 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = {\mathrm e}^{-2 x} \left (\cos \left (x \right )+1\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = x^{4} {\mathrm e}^{x}+x^{3} {\mathrm e}^{2 x}+x^{2} {\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.164 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = x \,{\mathrm e}^{-3 x} \sin \left (2 x \right )+x^{2} {\mathrm e}^{-2 x} \sin \left (3 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
74.813 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.099 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )^{3} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.048 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.925 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \tan \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.111 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = {\mathrm e}^{-2 x} \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.784 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{x} \tan \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.26 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = \frac {{\mathrm e}^{-3 x}}{x^{3}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x \,{\mathrm e}^{x} \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \csc \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.192 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{3} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.139 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \frac {1}{1+{\mathrm e}^{x}} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \frac {1}{{\mathrm e}^{2 x}+1} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.588 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )+1} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.454 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{x} \arcsin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \frac {{\mathrm e}^{-x}}{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.824 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 x y^{\prime }+10 y = 3 x^{4}+6 x^{3} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.574 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.099 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = \left (2+x \right )^{2} \] \(r = \frac {3}{\left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.499 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = x^{3} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.626 |
|
|
\[ {}x \left (-2+x \right ) y^{\prime \prime }-\left (x^{2}-2\right ) y^{\prime }+2 \left (-1+x \right ) y = 3 x^{2} \left (-2+x \right )^{2} {\mathrm e}^{x} \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.218 |
|
|
\[ {}\left (2 x +1\right ) \left (1+x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = \left (2 x +1\right )^{2} \] \(r = \frac {3}{\left (2 x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.093 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.478 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.734 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+3 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.205 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.919 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.893 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.219 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-4 x y^{\prime }+2 y = 0 \] \(r = \frac {4}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.035 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.791 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.997 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+10 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.097 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 4 x -6 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.065 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = 2 x^{3} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.416 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 4 \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.827 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 2 x \ln \left (x \right ) \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.044 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 4 \sin \left (\ln \left (x \right )\right ) \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
60.531 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-10 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.385 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.424 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.321 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 4 x -8 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.48 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y = -6 x^{3}+4 x^{2} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.759 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 10 x^{2} \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.566 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = 2 x^{3} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.29 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = \ln \left (x \right ) \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.129 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime }-\left (2+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 \left (2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.864 |
|
|
\[ {}\left (2 x -3\right )^{2} y^{\prime \prime }-6 \left (2 x -3\right ) y^{\prime }+12 y = 0 \] \(r = \frac {3}{\left (2 x -3\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.718 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }+2 x = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.742 |
|
|
\[ {}z^{\prime \prime }-4 z^{\prime }+13 z = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.068 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.57 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.386 |
|
|
\[ {}\theta ^{\prime \prime }+4 \theta = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.032 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.778 |
|
|
\[ {}2 z^{\prime \prime }+7 z^{\prime }-4 z = 0 \] \(r = {\frac {81}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.598 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.698 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+10 x = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.837 |
|
|
\[ {}4 x^{\prime \prime }-20 x^{\prime }+21 x = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.589 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.638 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.724 |
|
|
\[ {}y^{\prime \prime }+\omega ^{2} y = 0 \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.769 |
|
|
\[ {}x^{\prime \prime }-4 x = t^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.501 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime } = t^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.224 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }-2 x = 3 \,{\mathrm e}^{-t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.519 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }-2 x = {\mathrm e}^{t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+x = {\mathrm e}^{-t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}x^{\prime \prime }+\omega ^{2} x = \sin \left (\alpha t \right ) \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.94 |
|
|
\[ {}x^{\prime \prime }+\omega ^{2} x = \sin \left (\omega t \right ) \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.889 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.69 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t} \cos \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.966 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+10 x = {\mathrm e}^{-2 t} \cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.858 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+4 x = {\mathrm e}^{2 t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.577 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }-2 x = 12 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.707 |
|
|
\[ {}x^{\prime \prime }+4 x = 289 t \,{\mathrm e}^{t} \sin \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.481 |
|
|
\[ {}x^{\prime \prime }+\omega ^{2} x = \cos \left (\alpha t \right ) \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.436 |
|
|
\[ {}x^{\prime \prime }+\omega ^{2} x = \cos \left (\omega t \right ) \] \(r = -\omega ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.47 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = {\mathrm e}^{x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.483 |
|
|
\[ {}x^{\prime \prime }-x = \frac {1}{t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.52 |
|
|
\[ {}y^{\prime \prime }+4 y = \cot \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.353 |
|
|
\[ {}t^{2} x^{\prime \prime }-2 x = t^{3} \] \(r = \frac {2}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.075 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime } = \tan \left (t \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
12.559 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.365 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.976 |
|
|
\[ {}t^{2} x^{\prime \prime }-5 t x^{\prime }+10 x = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
4.056 |
|
|
\[ {}t^{2} x^{\prime \prime }+t x^{\prime }-x = 0 \] \(r = \frac {3}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.306 |
|
|
\[ {}x^{2} z^{\prime \prime }+3 x z^{\prime }+4 z = 0 \] \(r = -\frac {13}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
26.208 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.656 |
|
|
\[ {}4 t^{2} x^{\prime \prime }+8 t x^{\prime }+5 x = 0 \] \(r = -\frac {5}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.39 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y = 0 \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.652 |
|
|
\[ {}3 x^{2} z^{\prime \prime }+5 x z^{\prime }-z = 0 \] \(r = \frac {7}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.325 |
|
|
\[ {}t^{2} x^{\prime \prime }+3 t x^{\prime }+13 x = 0 \] \(r = -\frac {49}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
6.086 |
|
|
\[ {}a y^{\prime \prime }+\left (-a +b \right ) y^{\prime }+c y = 0 \] \(r = \frac {a^{2}-2 a b -4 a c +b^{2}}{4 a^{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.644 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+10 y = 100 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.149 |
|
|
\[ {}x^{\prime \prime }+x = \sin \left (t \right )-\cos \left (2 t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.955 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )^{3}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.08 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 2 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.027 |
|
|
\[ {}y^{\prime \prime }+y = \cosh \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.199 |
|
|
\[ {}x^{\prime \prime }-4 x^{\prime }+4 x = {\mathrm e}^{t}+{\mathrm e}^{2 t}+1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.817 |
|
|
\[ {}y^{\prime \prime }+y = 1-\frac {1}{\sin \left (x \right )} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.99 |
|
|
\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{r} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.149 |
|
|
\[ {}x^{\prime \prime }+9 x = t \sin \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.362 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \sinh \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.047 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = x \,{\mathrm e}^{x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.105 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-6 y = 1 \] \(r = \frac {6}{x^{2}-1}\) \(L = [1, 4, 6, 12]\) case used \(1\) poles order = \([1]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.853 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y = 2 \cos \left (\ln \left (1+x \right )\right ) \] \(r = -\frac {5}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
83.548 |
|
|
\[ {}x^{\prime \prime }+10 x^{\prime }+25 x = 2^{t}+t \,{\mathrm e}^{-5 t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (3 x \right ) \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.083 |
|
|
\[ {}x^{2} y^{\prime \prime }-y = \sin \left (x \right )^{2} \] \(r = \frac {1}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
9.76 |
|
|
\[ {}y^{\prime \prime } = y+x^{2} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.298 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }-2 y = 0 \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.284 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.327 |
|
|
\[ {}y^{\prime \prime }+2 x y^{\prime }+2 y = 0 \] \(r = x^{2}-1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.134 |
|
|
\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+4 x y = 2 x \] \(r = x \left (x^{3}-2\right )\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.698 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y = 1-2 x \] \(r = \frac {3 x +5}{\left (-4+4 x \right ) \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.099 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.53 |
|
|
\[ {}y^{\prime \prime }+x^{2} y^{\prime }+2 x y = 2 x \] \(r = \frac {x \left (x^{3}-4\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -4\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.692 |
|
|
\[ {}x y^{\prime \prime }+x^{2} y^{\prime }+2 x y = 0 \] \(r = \frac {x^{2}}{4}-\frac {3}{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.03 |
|
|
\[ {}x \ln \left (x \right ) y^{\prime \prime }+2 y^{\prime }-\frac {y}{x} = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.807 |
|
|
\[ {}y^{\prime \prime }+\frac {2 x y^{\prime }}{2 x -1}-\frac {4 x y}{\left (2 x -1\right )^{2}} = 0 \] \(r = \frac {x^{2}+4 x -1}{\left (2 x -1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.328 |
|
|
\[ {}\left (x^{2}+2 x \right ) y^{\prime \prime }+\left (x^{2}+x +10\right ) y^{\prime } = \left (25-6 x \right ) y \] \(r = \frac {x^{4}-22 x^{3}+75 x^{2}+180 x +60}{4 \left (x^{2}+2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.062 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{1+x}-\frac {\left (2+x \right ) y}{x^{2} \left (1+x \right )} = 0 \] \(r = \frac {3 x^{2}+12 x +8}{4 \left (x^{2}+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.566 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x^{2}+4 x -3\right ) y^{\prime }+8 x y = 0 \] \(r = \frac {4 x^{4}-16 x^{3}+24 x^{2}-12 x +3}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.878 |
|
|
\[ {}\frac {\left (x^{2}-x \right ) y^{\prime \prime }}{x}+\frac {\left (1+3 x \right ) y^{\prime }}{x}+\frac {y}{x} = 3 x \] \(r = \frac {-x^{2}+6 x +3}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.669 |
|
|
\[ {}y^{\prime \prime }+\left (5+2 x \right ) y^{\prime }+\left (4 x +8\right ) y = {\mathrm e}^{-2 x} \] \(r = -\frac {3}{4}+x^{2}+x\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.909 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = t^{7} \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.401 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.401 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-7 y = 4 \] \(r = {\frac {37}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.491 |
|
|
\[ {}3 y^{\prime \prime }+5 y^{\prime }-2 y = 3 t^{2} \] \(r = {\frac {49}{36}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.449 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x^{\frac {3}{2}} {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 \sec \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.069 |
|
|
\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y = x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.148 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+y = f \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.295 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2} = 0 \] \(r = \frac {4 x^{2}-4 x -3}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-y = 0 \] \(r = \frac {x^{2}+2 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.618 |
|
|
\[ {}y^{\prime \prime }+\alpha ^{2} y = 0 \] \(r = -\alpha ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.866 |
|
|
\[ {}y^{\prime \prime }-\alpha ^{2} y = 0 \] \(r = \alpha ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.759 |
|
|
\[ {}y^{\prime \prime }+\beta y^{\prime }+\gamma y = 0 \] \(r = \frac {\beta ^{2}}{4}-\gamma \) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.368 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }+k^{2} y = {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0 \] \(r = \frac {-4 a^{2} x^{2}+4 a^{2}-x^{2}-2}{4 \left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.716 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.6 |
|
|
\[ {}y^{\prime \prime } = a^{2} y \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.788 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} {\mathrm e}^{x} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.897 |
|
|
\[ {}y^{\prime \prime } = 9 y \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.745 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.826 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.66 |
|
|
\[ {}y^{\prime \prime }+12 y = 7 y^{\prime } \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.283 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.327 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.424 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-2 y = 0 \] \(r = {\frac {17}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.324 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.328 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.439 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = x \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.388 |
|
|
\[ {}s^{\prime \prime }-a^{2} s = t +1 \] \(r = a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.515 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = 8 \sin \left (2 x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime \prime }-y = 5 x +2 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.521 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+5 y = {\mathrm e}^{2 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.398 |
|
|
\[ {}y^{\prime \prime }+9 y = 6 \,{\mathrm e}^{3 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 2-6 x \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+3 y = {\mathrm e}^{-x} \cos \left (x \right ) \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.955 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.604 |
|
|
\[ {}y^{\prime \prime }+2 h y^{\prime }+n^{2} y = 0 \] \(r = h^{2}-n^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.816 |
|
|
\[ {}y^{\prime \prime }+n^{2} y = h \sin \left (r x \right ) \] \(r = -n^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.307 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = \sin \left (x \right ) \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.477 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\cos \left (2 x \right )^{\frac {3}{2}}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.435 |
|
|
\[ {}y^{\prime \prime }-4 y = {\mathrm e}^{2 x} \sin \left (2 x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.616 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.733 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.287 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.987 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.102 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.276 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.337 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime } = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.579 |
|
|
\[ {}x^{2} y^{\prime \prime }+6 x y^{\prime }+4 y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.641 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.111 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.291 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.462 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.645 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.969 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.894 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.881 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.908 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.438 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
N/A |
1.063 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }+4 y = x \] \(r = -{\frac {11}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.704 |
|
|
\[ {}x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime } = x^{2} \] \(r = \frac {27-12 x}{4 \left (x^{2}-3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.86 |
|
|
\[ {}x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime } = x^{2} \] \(r = \frac {27-12 x}{4 \left (x^{2}-3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 3\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.83 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.774 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.88 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.391 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.7 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.388 |
|
|
\[ {}y^{\prime \prime }-4 y = 31 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.849 |
|
|
\[ {}y^{\prime \prime }+9 y = 27 x +18 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.924 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = -3 x -\frac {3}{x} \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.217 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }-3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.296 |
|
|
\[ {}y^{\prime \prime }+\alpha y = 0 \] \(r = -\alpha \) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.414 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }-7 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.358 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime }+2 y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.259 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = {\mathrm e}^{4 t} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.419 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = 2 \,{\mathrm e}^{-3 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.429 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \,{\mathrm e}^{3 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = {\mathrm e}^{-t} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.695 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = -3 \,{\mathrm e}^{-2 t} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = {\mathrm e}^{-2 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.519 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+4 y = {\mathrm e}^{4 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 4 \,{\mathrm e}^{-3 t} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = {\mathrm e}^{-t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.659 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+12 y = 3 \,{\mathrm e}^{-t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = -3 \,{\mathrm e}^{-2 t} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.725 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = {\mathrm e}^{-2 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = {\mathrm e}^{-\frac {t}{2}} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = {\mathrm e}^{-2 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.631 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = {\mathrm e}^{-4 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = {\mathrm e}^{-\frac {t}{2}} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.03 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = {\mathrm e}^{-2 t} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.856 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = {\mathrm e}^{-4 t} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.805 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.487 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+4 y = 5 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.639 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 2 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.617 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 10 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.859 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+6 y = -8 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.246 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{-t} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.798 |
|
|
\[ {}y^{\prime \prime }+4 y = 2 \,{\mathrm e}^{-2 t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.794 |
|
|
\[ {}y^{\prime \prime }+2 y = -3 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.763 |
|
|
\[ {}y^{\prime \prime }+4 y = {\mathrm e}^{t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.709 |
|
|
\[ {}y^{\prime \prime }+9 y = 6 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.707 |
|
|
\[ {}y^{\prime \prime }+2 y = -{\mathrm e}^{t} \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime }+4 y = -3 t^{2}+2 t +3 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.888 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 3 t +2 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.078 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 3 t +2 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.102 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = t^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.691 |
|
|
\[ {}y^{\prime \prime }+4 y = t -\frac {1}{20} t^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.879 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 4+{\mathrm e}^{-t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{-t}-4 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.706 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = 2 t +{\mathrm e}^{-t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.717 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = 2 t +{\mathrm e}^{t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.714 |
|
|
\[ {}y^{\prime \prime }+4 y = t +{\mathrm e}^{-t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}y^{\prime \prime }+4 y = 6+t^{2}+{\mathrm e}^{t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.958 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.509 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 5 \cos \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.452 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \sin \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 2 \sin \left (t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.452 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = \cos \left (t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = -4 \cos \left (3 t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.553 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = 3 \cos \left (2 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.849 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = -\cos \left (5 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = -3 \sin \left (2 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \cos \left (3 t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.707 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = \cos \left (t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = 2 \cos \left (3 t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+20 y = -3 \sin \left (2 t \right ) \] \(r = -11\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.94 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 2 \cos \left (2 t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+y = \cos \left (3 t \right ) \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.755 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = 3+2 \cos \left (2 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = {\mathrm e}^{-t} \cos \left (t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.785 |
|
|
\[ {}y^{\prime \prime }+9 y = \cos \left (t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.794 |
|
|
\[ {}y^{\prime \prime }+9 y = 5 \sin \left (2 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+4 y = -\cos \left (\frac {t}{2}\right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.859 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 \cos \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.643 |
|
|
\[ {}y^{\prime \prime }+9 y = 2 \cos \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime } = \frac {1+x}{-1+x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.842 |
|
|
\[ {}x^{2} y^{\prime \prime } = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+8 y = {\mathrm e}^{-x^{2}} \] \(r = -{\frac {23}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.121 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime } = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.748 |
|
|
\[ {}y^{\prime \prime } = \sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.073 |
|
|
\[ {}y^{\prime \prime }-3 = x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x y^{\prime \prime }+2 = \sqrt {x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.524 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime } = 18 x^{2} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x y^{\prime \prime } = 2 y^{\prime } \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.237 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 8 \,{\mathrm e}^{2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.638 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }-2 x^{2} y^{\prime } \] \(r = \frac {4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.319 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \] \(r = \frac {1}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.174 |
|
|
\[ {}y^{\prime \prime } = 2 y^{\prime }-6 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.451 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 9 \,{\mathrm e}^{-3 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.634 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = 6 x^{5} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.281 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 9 \,{\mathrm e}^{-3 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.542 |
|
|
\[ {}x y^{\prime \prime }+4 y^{\prime } = 18 x^{2} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.672 |
|
|
\[ {}x y^{\prime \prime } = 2 y^{\prime } \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.525 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.114 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 8 \,{\mathrm e}^{2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.023 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime } = 6 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.515 |
|
|
\[ {}y^{\prime \prime } = 2 y^{\prime }-5 y+30 \,{\mathrm e}^{3 x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.893 |
|
|
\[ {}y^{\prime \prime }+4 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.091 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.599 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.513 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.624 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.694 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.521 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.474 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
3.0 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.409 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
N/A |
1.587 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
N/A |
1.619 |
|
|
\[ {}y^{\prime \prime }-4 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.792 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.496 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+9 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.049 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.264 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-24 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime \prime }-25 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.19 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.973 |
|
|
\[ {}4 y^{\prime \prime }-y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.026 |
|
|
\[ {}3 y^{\prime \prime }+7 y^{\prime }-6 y = 0 \] \(r = {\frac {121}{36}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.285 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.525 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.502 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+15 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.497 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.712 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.554 |
|
|
\[ {}y^{\prime \prime }-9 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.297 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.32 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.312 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.331 |
|
|
\[ {}25 y^{\prime \prime }-10 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.338 |
|
|
\[ {}16 y^{\prime \prime }-24 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.348 |
|
|
\[ {}9 y^{\prime \prime }+12 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.342 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.618 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.664 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.621 |
|
|
|
|||||
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.694 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.648 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.699 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.431 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.479 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.311 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+29 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.521 |
|
|
\[ {}9 y^{\prime \prime }+18 y^{\prime }+10 y = 0 \] \(r = -{\frac {1}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.526 |
|
|
\[ {}4 y^{\prime \prime }+y = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.109 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.506 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.408 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.925 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.753 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.563 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.59 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (\frac {1}{4}+4 \pi ^{2}\right ) y = 0 \] \(r = -4 \pi ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.89 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (\frac {1}{4}+4 \pi ^{2}\right ) y = 0 \] \(r = -4 \pi ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.817 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.617 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.531 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime } = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.826 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.974 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.55 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.57 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.464 |
|
|
\[ {}x^{2} y^{\prime \prime }-19 x y^{\prime }+100 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.618 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+29 y = 0 \] \(r = -\frac {81}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.948 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+10 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.757 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+29 y = 0 \] \(r = -\frac {101}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.694 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.405 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.679 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+37 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.568 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.83 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-25 y = 0 \] \(r = \frac {99}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.443 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+5 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.579 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-7 x y^{\prime }+3 y = 0 \] \(r = \frac {55}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.578 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-10 y = 0 \] \(r = \frac {12}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.962 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }-y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.485 |
|
|
\[ {}x^{2} y^{\prime \prime }-11 x y^{\prime }+36 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.323 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.377 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.802 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+13 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.096 |
|
|
\[ {}y^{\prime \prime }+4 y = 24 \,{\mathrm e}^{2 x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.781 |
|
|
\[ {}y^{\prime \prime }+4 y = 24 \,{\mathrm e}^{2 x} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.721 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-8 y = 8 x^{2}-3 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-8 y = 8 x^{2}-3 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.677 |
|
|
\[ {}y^{\prime \prime }-9 y = 36 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.493 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -6 \,{\mathrm e}^{4 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.704 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 7 \,{\mathrm e}^{5 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 169 \sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.215 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 10 x +12 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.763 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = {\mathrm e}^{4 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.427 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = {\mathrm e}^{5 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.499 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -18 \,{\mathrm e}^{4 x}+14 \,{\mathrm e}^{5 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 35 \,{\mathrm e}^{5 x}+12 \,{\mathrm e}^{4 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.65 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.341 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.326 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 22 x +24 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.988 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = x \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.911 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 1 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.905 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 4 x^{2}+2 x +3 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.025 |
|
|
\[ {}y^{\prime \prime }+9 y = 52 \,{\mathrm e}^{2 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.723 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 27 \,{\mathrm e}^{6 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.551 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = 30 \,{\mathrm e}^{-4 x} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = {\mathrm e}^{\frac {x}{2}} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.729 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -5 \,{\mathrm e}^{3 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime }+9 y = 10 \cos \left (2 x \right )+15 \sin \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.36 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 25 \sin \left (6 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.85 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 26 \cos \left (\frac {x}{3}\right )-12 \sin \left (\frac {x}{3}\right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.344 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = \cos \left (x \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -4 \cos \left (x \right )+7 \sin \left (x \right ) \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.979 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -200 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = x^{3} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.469 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 18 x^{2}+3 x +4 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.549 |
|
|
\[ {}y^{\prime \prime }+9 y = 9 x^{4}-9 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.703 |
|
|
\[ {}y^{\prime \prime }+9 y = x^{3} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.901 |
|
|
\[ {}y^{\prime \prime }+9 y = 25 x \cos \left (2 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.119 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.738 |
|
|
\[ {}y^{\prime \prime }+9 y = 54 x^{2} {\mathrm e}^{3 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime } = 6 x \,{\mathrm e}^{x} \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
2.779 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \left (-6 x -8\right ) \cos \left (2 x \right )+\left (8 x -11\right ) \sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.388 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \left (12 x -4\right ) {\mathrm e}^{-5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }+9 y = 39 \,{\mathrm e}^{2 x} x \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.919 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = -3 \,{\mathrm e}^{-2 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.458 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 20 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.543 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = x^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.74 |
|
|
\[ {}y^{\prime \prime }+9 y = 3 \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.773 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 10 \,{\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.523 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = \left (72 x^{2}-1\right ) {\mathrm e}^{2 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }-10 y = 4 x \,{\mathrm e}^{6 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.478 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 6 \,{\mathrm e}^{5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.529 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 6 \,{\mathrm e}^{-5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 24 \sin \left (3 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 8 \,{\mathrm e}^{-3 x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{-x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.724 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 100 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.563 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{-x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 10 x^{2}+4 x +8 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.652 |
|
|
\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime \prime }+y = 6 \cos \left (x \right )-3 \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.051 |
|
|
\[ {}y^{\prime \prime }+y = 6 \cos \left (2 x \right )-3 \sin \left (2 x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.216 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = x^{3} {\mathrm e}^{-x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.536 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = x^{3} {\mathrm e}^{2 x} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.343 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{-7 x}+2 \,{\mathrm e}^{-7 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.539 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.455 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 4 \,{\mathrm e}^{-8 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.444 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 4 \,{\mathrm e}^{3 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.446 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{3 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.484 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} \cos \left (2 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.149 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = x^{2} {\mathrm e}^{3 x} \sin \left (2 x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.245 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = {\mathrm e}^{4 x} \sin \left (2 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.263 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = {\mathrm e}^{2 x} \sin \left (4 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.74 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = x^{3} \sin \left (4 x \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.003 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 3 x^{2} {\mathrm e}^{5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 3 x^{4} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.602 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 27 \,{\mathrm e}^{6 x}+25 \sin \left (6 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.159 |
|
|
\[ {}y^{\prime \prime }+9 y = 25 x \cos \left (2 x \right )+3 \sin \left (3 x \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.902 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 5 \sin \left (x \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.822 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 20 \sinh \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.024 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = \frac {5}{x^{3}} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.225 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = \frac {50}{x^{3}} \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.514 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 85 \cos \left (2 \ln \left (x \right )\right ) \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
14.775 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 15 \cos \left (3 \ln \left (x \right )\right )-10 \sin \left (3 \ln \left (x \right )\right ) \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.625 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-7 x y^{\prime }+3 y = 4 x^{3} \] \(r = \frac {55}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.026 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = \frac {10}{x} \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.096 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 6 x^{3} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.621 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 64 \ln \left (x \right ) x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.189 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 3 \sqrt {x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.726 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.758 |
|
|
\[ {}y^{\prime \prime }+4 y = \csc \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 6 \,{\mathrm e}^{3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.468 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \left (24 x^{2}+2\right ) {\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.661 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = \frac {{\mathrm e}^{-2 x}}{x^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.649 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \sqrt {x} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.602 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 12 x^{3} \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.957 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{2} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.657 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = \ln \left (x \right ) \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.113 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = \frac {1}{-2+x} \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.904 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }-4 x^{3} y = x^{3} {\mathrm e}^{x^{2}} \] \(r = \frac {16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.109 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 8 \,{\mathrm e}^{2 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.189 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] \(r = \frac {x^{2}+4 x +6}{4 \left (1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.115 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y = \frac {10}{x} \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.388 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-6 y = 12 \,{\mathrm e}^{2 x} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.742 |
|
|
\[ {}y^{\prime \prime }+36 y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.648 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.351 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0 \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.517 |
|
|
\[ {}y^{\prime \prime }-36 y = 0 \] \(r = 36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.254 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+16 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.699 |
|
|
\[ {}2 x y^{\prime \prime }+y^{\prime } = \sqrt {x} \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.704 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}y^{\prime \prime }+3 y = 0 \] \(r = -3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.802 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.795 |
|
|
\[ {}x^{2} y^{\prime \prime }+\frac {5 y}{2} = 0 \] \(r = -\frac {5}{2 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.704 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 y = 0 \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.476 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+25 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.576 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+9 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.559 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+25 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.513 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-30 y = 0 \] \(r = \frac {30}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-30 y = 0 \] \(r = {\frac {121}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.28 |
|
|
\[ {}16 y^{\prime \prime }-8 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.347 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.596 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-3 x y^{\prime }+2 y = 0 \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.631 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.632 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }+3 = 0 \] \(r = {\frac {49}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.789 |
|
|
\[ {}y^{\prime \prime }+20 y^{\prime }+100 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}x y^{\prime \prime } = 3 y^{\prime } \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.601 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime } = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.03 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 98 x^{2} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.49 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 25 \sin \left (3 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.83 |
|
|
\[ {}y^{\prime \prime }-9 y^{\prime }+14 y = 576 \,{\mathrm e}^{-x} x^{2} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 81 \,{\mathrm e}^{3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 3 \sqrt {x} \] \(r = \frac {35}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.888 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+36 y = 3 x \,{\mathrm e}^{6 x}-2 \,{\mathrm e}^{6 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }+36 y = 6 \sec \left (6 x \right ) \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.027 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 18 \ln \left (x \right ) \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.647 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 10 \,{\mathrm e}^{-3 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.547 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }-2 y = 10 x^{2} \] \(r = \frac {21}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.115 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 2 \cos \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.775 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+2 y = 6 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.336 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \frac {1}{x^{2}+1} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.786 |
|
|
\[ {}4 y^{\prime \prime }-12 y^{\prime }+9 y = x \,{\mathrm e}^{\frac {3 x}{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.587 |
|
|
\[ {}3 y^{\prime \prime }+8 y^{\prime }-3 y = 123 x \sin \left (3 x \right ) \] \(r = {\frac {25}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.035 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{\left (1+x \right )^{2}} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.776 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \frac {1}{x} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.302 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = x^{3} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.791 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+2 y = 0 \] \(r = -\frac {9}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.947 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.431 |
|
|
\[ {}y^{\prime \prime }+9 y^{\prime } = 0 \] \(r = {\frac {81}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.816 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }-10 x = 0 \] \(r = 11\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.527 |
|
|
\[ {}x^{\prime \prime }+x = t \cos \left (t \right )-\cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.711 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+40 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.954 |
|
|
\[ {}x^{2} y^{\prime \prime }-12 x y^{\prime }+42 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.977 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+5 y = 0 \] \(r = -\frac {17}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.825 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.905 |
|
|
\[ {}y^{\prime \prime }+9 y^{\prime } = 0 \] \(r = {\frac {81}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.472 |
|
|
\[ {}t^{2} y^{\prime \prime }-12 t y^{\prime }+42 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
4.648 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+5 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
5.391 |
|
|
\[ {}16 y^{\prime \prime }+24 y^{\prime }+153 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.883 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }+4 y^{\prime }-5 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.43 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+45 y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.921 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-16 y = 0 \] \(r = \frac {67}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
5.091 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+2 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.804 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.148 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = 2 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.668 |
|
|
\[ {}y^{\prime \prime }+4 y = t \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.134 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.826 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{t}+\frac {y}{t^{2}} = \frac {1}{t} \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.964 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.292 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.381 |
|
|
\[ {}2 t^{2} y^{\prime \prime }-3 t y^{\prime }-3 y = 0 \] \(r = \frac {45}{16 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.671 |
|
|
\[ {}y^{\prime \prime }+9 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.941 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.608 |
|
|
\[ {}y^{\prime \prime }+9 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.305 |
|
|
\[ {}3 t^{2} y^{\prime \prime }-5 t y^{\prime }-3 y = 0 \] \(r = \frac {91}{36 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.791 |
|
|
\[ {}t^{2} y^{\prime \prime }+7 t y^{\prime }-7 y = 0 \] \(r = \frac {63}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
4.243 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.036 |
|
|
\[ {}y^{\prime \prime }+10 y^{\prime }+24 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.322 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.914 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+18 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.708 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }-y = 0 \] \(r = \frac {3}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.97 |
|
|
\[ {}a y^{\prime \prime }+b y^{\prime }+c y = 0 \] \(r = \frac {-4 a c +b^{2}}{4 a^{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.607 |
|
|
\[ {}t^{2} y^{\prime \prime }+a t y^{\prime }+b y = 0 \] \(r = \frac {a^{2}-2 a -4 b}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.349 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.84 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-12 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.328 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.27 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+12 y = 0 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.325 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+y = 0 \] \(r = {\frac {21}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.386 |
|
|
\[ {}8 y^{\prime \prime }+6 y^{\prime }+y = 0 \] \(r = {\frac {1}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.342 |
|
|
\[ {}4 y^{\prime \prime }+9 y = 0 \] \(r = -{\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.474 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.194 |
|
|
\[ {}y^{\prime \prime }+8 y = 0 \] \(r = -8\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.447 |
|
|
\[ {}y^{\prime \prime }+7 y = 0 \] \(r = -7\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.267 |
|
|
\[ {}4 y^{\prime \prime }+21 y^{\prime }+5 y = 0 \] \(r = {\frac {361}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.344 |
|
|
\[ {}7 y^{\prime \prime }+4 y^{\prime }-3 y = 0 \] \(r = {\frac {25}{49}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.339 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.391 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.372 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.337 |
|
|
\[ {}3 y^{\prime \prime }-y^{\prime } = 0 \] \(r = {\frac {1}{36}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.742 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-12 y = 0 \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.576 |
|
|
\[ {}2 y^{\prime \prime }-7 y^{\prime }-4 y = 0 \] \(r = {\frac {81}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.596 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.545 |
|
|
\[ {}y^{\prime \prime }+36 y = 0 \] \(r = -36\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
8.265 |
|
|
\[ {}y^{\prime \prime }+100 y = 0 \] \(r = -100\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
14.04 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.761 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.944 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.9 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.975 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.251 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+y = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.296 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-y = 0 \] \(r = {\frac {5}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.0 |
|
|
\[ {}6 y^{\prime \prime }+5 y^{\prime }+y = 0 \] \(r = {\frac {1}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.339 |
|
|
\[ {}9 y^{\prime \prime }+6 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.399 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.379 |
|
|
\[ {}3 t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y = 0 \] \(r = -\frac {2}{9 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.668 |
|
|
\[ {}t^{2} y^{\prime \prime }-t y^{\prime }+y = 0 \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.598 |
|
|
\[ {}a y^{\prime \prime }+2 b y^{\prime }+c y = 0 \] \(r = \frac {-a c +b^{2}}{a^{2}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.615 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+2 y = 0 \] \(r = 7\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.372 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.318 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }-16 y = 0 \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.329 |
|
|
\[ {}y^{\prime \prime }-16 y = 0 \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.61 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.539 |
|
|
\[ {}y^{\prime \prime }+y = 8 \,{\mathrm e}^{2 t} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.694 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = -{\mathrm e}^{-9 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 2 \,{\mathrm e}^{3 t} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.553 |
|
|
\[ {}y^{\prime \prime }-y = 2 t -4 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.496 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = t^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 3-4 t \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.054 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (2 t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.089 |
|
|
\[ {}y^{\prime \prime }+4 y = 4 \cos \left (t \right )-\sin \left (t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.691 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (2 t \right )+t \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.235 |
|
|
\[ {}y^{\prime \prime }+4 y = 3 t \,{\mathrm e}^{-t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.744 |
|
|
\[ {}y^{\prime \prime } = 3 t^{4}-2 t \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.902 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+13 y = 2 t \,{\mathrm e}^{-2 t} \sin \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.512 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = -1 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.485 |
|
|
\[ {}5 y^{\prime \prime }+y^{\prime }-4 y = -3 \] \(r = {\frac {81}{100}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 32 t \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.509 |
|
|
\[ {}16 y^{\prime \prime }-8 y^{\prime }-15 y = 75 t \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+26 y = -338 t \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.02 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = -32 t^{2} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}8 y^{\prime \prime }+6 y^{\prime }+y = 5 t^{2} \] \(r = {\frac {1}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.559 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = -256 t^{3} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.552 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 52 \sin \left (3 t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.997 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 25 \sin \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.119 |
|
|
\[ {}y^{\prime \prime }-9 y = 54 t \sin \left (2 t \right ) \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.024 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = -78 \cos \left (3 t \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.682 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = -32 t^{2} \cos \left (2 t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.672 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-20 y = -2 \,{\mathrm e}^{t} \] \(r = {\frac {81}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.52 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }-5 y = -648 t^{2} {\mathrm e}^{5 t} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.655 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+12 y = -2 t^{3} {\mathrm e}^{4 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.604 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = 8 \,{\mathrm e}^{4 t}-4 \,{\mathrm e}^{-4 t} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.337 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = t^{2}-{\mathrm e}^{3 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.58 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = -24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.517 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = t^{2}-{\mathrm e}^{3 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.22 |
|
|
\[ {}y^{\prime \prime } = t^{2}+{\mathrm e}^{t}+\sin \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
3.571 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 18 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.643 |
|
|
\[ {}y^{\prime \prime }-y = 4 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.685 |
|
|
\[ {}y^{\prime \prime }-4 y = 32 t \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.755 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-3 y = -2 \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.733 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = 3 t \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.772 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+16 y = 4 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.926 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = t \,{\mathrm e}^{-t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.832 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+25 y = -1 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.244 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = -{\mathrm e}^{3 t}-2 t \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.026 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = -3 t -4 t^{2} {\mathrm e}^{2 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.357 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = 2 t^{2} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.508 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = -24 t -6-4 t \,{\mathrm e}^{-4 t}+{\mathrm e}^{-4 t} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.844 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = {\mathrm e}^{-3 t}-{\mathrm e}^{3 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.079 |
|
|
\[ {}y^{\prime \prime }+9 y = \left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
7.708 |
|
|
\[ {}y^{\prime \prime }+9 \pi ^{2} y = \left \{\begin {array}{cc} 2 t & 0\le t <\pi \\ 2 t -2 \pi & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \] \(r = -9 \pi ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
29.137 |
|
|
\[ {}y^{\prime \prime }+4 y = \left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 10 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.716 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = f \left (t \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.669 |
|
|
\[ {}x^{\prime \prime }+9 x = \sin \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.256 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+37 y = \cos \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.283 |
|
|
\[ {}y^{\prime \prime }+4 y = 1 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.166 |
|
|
\[ {}y^{\prime \prime }+16 y^{\prime } = t \] \(r = 64\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.14 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = {\mathrm e}^{3 t} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }+16 y = 2 \cos \left (4 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+20 y = 2 t \,{\mathrm e}^{-2 t} \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }+\frac {y}{4} = \sec \left (\frac {t}{2}\right )+\csc \left (\frac {t}{2}\right ) \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.672 |
|
|
\[ {}y^{\prime \prime }+16 y = \csc \left (4 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime }+16 y = \cot \left (4 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.523 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+50 y = {\mathrm e}^{-t} \csc \left (7 t \right ) \] \(r = -49\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.307 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+25 y = {\mathrm e}^{-3 t} \left (\sec \left (4 t \right )+\csc \left (4 t \right )\right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.216 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+26 y = {\mathrm e}^{t} \left (\sec \left (5 t \right )+\csc \left (5 t \right )\right ) \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.49 |
|
|
\[ {}y^{\prime \prime }+12 y^{\prime }+37 y = {\mathrm e}^{-6 t} \csc \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.991 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+34 y = {\mathrm e}^{3 t} \tan \left (5 t \right ) \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.489 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+34 y = {\mathrm e}^{5 t} \cot \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.442 |
|
|
\[ {}y^{\prime \prime }-12 y^{\prime }+37 y = {\mathrm e}^{6 t} \sec \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+17 y = {\mathrm e}^{4 t} \sec \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.895 |
|
|
\[ {}y^{\prime \prime }-9 y = \frac {1}{1+{\mathrm e}^{3 t}} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.72 |
|
|
\[ {}y^{\prime \prime }-25 y = \frac {1}{1-{\mathrm e}^{5 t}} \] \(r = 25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.772 |
|
|
\[ {}y^{\prime \prime }-y = 2 \sinh \left (t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.128 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{t}}{t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.651 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \frac {{\mathrm e}^{2 t}}{t^{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.666 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+16 y = \frac {{\mathrm e}^{-4 t}}{t^{4}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = \frac {{\mathrm e}^{-3 t}}{t} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = {\mathrm e}^{-3 t} \ln \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.705 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left ({\mathrm e}^{t}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.827 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = {\mathrm e}^{-2 t} \sqrt {-t^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.816 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{t} \sqrt {-t^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.732 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = {\mathrm e}^{5 t} \ln \left (2 t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.81 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{2 t} \arctan \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.765 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime }+16 y = \frac {{\mathrm e}^{-4 t}}{t^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.68 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (\frac {t}{2}\right )+\csc \left (\frac {t}{2}\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.872 |
|
|
\[ {}y^{\prime \prime }+9 y = \tan \left (3 t \right )^{2} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.764 |
|
|
\[ {}y^{\prime \prime }+9 y = \sec \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.02 |
|
|
\[ {}y^{\prime \prime }+9 y = \tan \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.251 |
|
|
\[ {}y^{\prime \prime }+4 y = \tan \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.305 |
|
|
\[ {}y^{\prime \prime }+16 y = \tan \left (2 t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.408 |
|
|
\[ {}y^{\prime \prime }+4 y = \tan \left (t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.132 |
|
|
\[ {}y^{\prime \prime }+9 y = \sec \left (3 t \right ) \tan \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.364 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (2 t \right ) \tan \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }+9 y = \frac {\csc \left (3 t \right )}{2} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.352 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (2 t \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.241 |
|
|
\[ {}y^{\prime \prime }-16 y = 16 t \,{\mathrm e}^{-4 t} \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.909 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (t \right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.585 |
|
|
\[ {}y^{\prime \prime }+4 y = \sec \left (2 t \right )+\tan \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.411 |
|
|
\[ {}y^{\prime \prime }+9 y = \csc \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.619 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 65 \cos \left (2 t \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.66 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = \ln \left (t \right ) \] \(r = -\frac {1}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
5.294 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+4 y = t \] \(r = -\frac {17}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.547 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }-6 y = 2 \ln \left (t \right ) \] \(r = \frac {12}{t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.653 |
|
|
\[ {}4 y^{\prime \prime }+4 y^{\prime }+y = {\mathrm e}^{-\frac {t}{2}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }+4 y = f \left (t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.829 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y = t^{3}+2 t \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.562 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = -t \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.221 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 16 t^{\frac {3}{2}} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
1.503 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 16 t^{\frac {3}{2}} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.326 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-8 x y^{\prime }+5 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.299 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-4 x y^{\prime }+2 y = 0 \] \(r = \frac {4}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.232 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-8 x y^{\prime }+8 y = 0 \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.76 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-7 x y^{\prime }+7 y = 0 \] \(r = \frac {21}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.839 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+17 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.803 |
|
|
\[ {}9 x^{2} y^{\prime \prime }-9 x y^{\prime }+10 y = 0 \] \(r = -\frac {13}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.359 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-2 x y^{\prime }+20 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.325 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+10 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.247 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+8 x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.172 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.613 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.172 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+9 y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.17 |
|
|
\[ {}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = \frac {1}{x^{5}} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.681 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+9 y = x^{3} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.467 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = \frac {1}{x^{2}} \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
9.394 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = \frac {1}{x^{2}} \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
9.445 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-6 y = 2 x \] \(r = \frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.912 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-16 y = \ln \left (x \right ) \] \(r = \frac {63}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.49 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 8 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.369 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+36 y = x^{2} \] \(r = -\frac {145}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
7.129 |
|
|
\[ {}3 x^{2} y^{\prime \prime }-4 x y^{\prime }+2 y = 0 \] \(r = \frac {4}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
4.274 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-7 x y^{\prime }+7 y = 0 \] \(r = \frac {21}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
5.136 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.483 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+2 y = 0 \] \(r = -\frac {9}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
4.565 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = \frac {1}{x^{2}} \] \(r = \frac {5}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.778 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.766 |
|
|
|
|||||
|
\[ {}4 x^{2} y^{\prime \prime }+y = x^{3} \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.23 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+27 x y^{\prime }+10 y = \frac {1}{x} \] \(r = -\frac {13}{36 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
11.657 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.231 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.191 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.844 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.355 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = \arctan \left (x \right ) \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.633 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = 0 \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.166 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = \arctan \left (x \right ) \] \(r = -\frac {3}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.023 |
|
|
\[ {}\left (x^{4}-1\right ) y^{\prime \prime }+\left (x^{3}-x \right ) y^{\prime }+\left (x^{2}-1\right ) y = 0 \] \(r = \frac {-5 x^{2}-2}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.791 |
|
|
\[ {}\left (x^{4}-1\right ) y^{\prime \prime }+\left (x^{3}-x \right ) y^{\prime }+\left (4 x^{2}-4\right ) y = 0 \] \(r = \frac {-17 x^{2}-14}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.724 |
|
|
\[ {}\left (x^{4}-1\right ) y^{\prime \prime }+\left (x^{3}-x \right ) y^{\prime }+\left (x^{2}-1\right ) y = 0 \] \(r = \frac {-5 x^{2}-2}{4 \left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.668 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
4.118 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = x^{2} \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.156 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+4 y = 0 \] \(r = -\frac {17}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.634 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.135 |
|
|
\[ {}6 x^{2} y^{\prime \prime }+5 x y^{\prime }-y = 0 \] \(r = -\frac {11}{144 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.089 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.341 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.333 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.562 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+10 y = 0 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.357 |
|
|
\[ {}6 y^{\prime \prime }+5 y^{\prime }-4 y = 0 \] \(r = {\frac {121}{144}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+34 y = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.772 |
|
|
\[ {}2 y^{\prime \prime }-5 y^{\prime }+2 y = 0 \] \(r = {\frac {9}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.362 |
|
|
\[ {}15 y^{\prime \prime }-11 y^{\prime }+2 y = 0 \] \(r = {\frac {1}{900}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.365 |
|
|
\[ {}20 y^{\prime \prime }+y^{\prime }-y = 0 \] \(r = {\frac {81}{1600}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.366 |
|
|
\[ {}12 y^{\prime \prime }+8 y^{\prime }+y = 0 \] \(r = {\frac {1}{36}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.359 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = -t \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.536 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime } = 5 t^{2} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.464 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = -3 \sin \left (t \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.441 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 3 \sin \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.237 |
|
|
\[ {}y^{\prime \prime }-9 y = \frac {1}{1+{\mathrm e}^{3 t}} \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.692 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime } = \frac {1}{{\mathrm e}^{2 t}+1} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.274 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = -4 \,{\mathrm e}^{-2 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.534 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 3 \,{\mathrm e}^{-2 t} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.025 |
|
|
\[ {}y^{\prime \prime }+9 y^{\prime }+20 y = -2 t \,{\mathrm e}^{t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.578 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime }+12 y = 3 t^{2} {\mathrm e}^{-4 t} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.593 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.646 |
|
|
\[ {}y^{\prime \prime }+10 y^{\prime }+16 y = 0 \] \(r = 9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.624 |
|
|
\[ {}y^{\prime \prime }+16 y = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.389 |
|
|
\[ {}y^{\prime \prime }+25 y = 0 \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.296 |
|
|
\[ {}y^{\prime \prime }-4 y = t \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.78 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = {\mathrm e}^{t} \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.86 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (3 t \right ) \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.433 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.142 |
|
|
\[ {}y^{\prime \prime }+4 y = \tan \left (2 t \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.158 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.833 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = \frac {{\mathrm e}^{4 t}}{t^{3}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = \frac {{\mathrm e}^{4 t}}{t^{3}} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.809 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{t} \ln \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.706 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{t} \ln \left (t \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.311 |
|
|
\[ {}y^{\prime \prime }-2 t y^{\prime }+t^{2} y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.487 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }-4 y = 0 \] \(r = {\frac {25}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.351 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.41 |
|
|
\[ {}t^{2} y^{\prime \prime }-5 t y^{\prime }+5 y = 0 \] \(r = \frac {15}{4 t^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.047 |
|
|
\[ {}x^{2} y^{\prime \prime }+7 x y^{\prime }+8 y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.328 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.554 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.039 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+y = 0 \] \(r = -\frac {3}{16 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
6.456 |
|
|
\[ {}5 x^{2} y^{\prime \prime }-x y^{\prime }+2 y = 0 \] \(r = -\frac {29}{100 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.429 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+25 y = 0 \] \(r = -\frac {37}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
2.359 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 8 x \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.712 |
|
|
\[ {}4 x^{\prime \prime }+9 x = 0 \] \(r = -{\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
8.311 |
|
|
\[ {}9 x^{\prime \prime }+4 x = 0 \] \(r = -{\frac {4}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
7.336 |
|
|
\[ {}x^{\prime \prime }+64 x = 0 \] \(r = -64\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
16.018 |
|
|
\[ {}x^{\prime \prime }+100 x = 0 \] \(r = -100\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
11.646 |
|
|
\[ {}x^{\prime \prime }+x = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.997 |
|
|
\[ {}x^{\prime \prime }+4 x = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.296 |
|
|
\[ {}x^{\prime \prime }+16 x = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
8.678 |
|
|
\[ {}x^{\prime \prime }+256 x = 0 \] \(r = -256\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
27.919 |
|
|
\[ {}x^{\prime \prime }+9 x = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.13 |
|
|
\[ {}10 x^{\prime \prime }+\frac {x}{10} = 0 \] \(r = -{\frac {1}{100}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.366 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+3 x = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.626 |
|
|
\[ {}\frac {x^{\prime \prime }}{32}+2 x^{\prime }+x = 0 \] \(r = 992\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.11 |
|
|
\[ {}\frac {x^{\prime \prime }}{4}+2 x^{\prime }+x = 0 \] \(r = 12\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.369 |
|
|
\[ {}4 x^{\prime \prime }+2 x^{\prime }+8 x = 0 \] \(r = -{\frac {31}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.602 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+13 x = 0 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.007 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+20 x = 0 \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.061 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
6.295 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
10.138 |
|
|
\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} t & 0\le t <1 \\ 2-t & 1\le t <2 \\ 0 & 2\le t \end {array}\right . \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
8.455 |
|
|
\[ {}x^{\prime \prime }+4 x^{\prime }+13 x = \left \{\begin {array}{cc} 1 & 0\le t <\pi \\ 1-t & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
30.017 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.033 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (t \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (\frac {9 t}{10}\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.942 |
|
|
\[ {}x^{\prime \prime }+x = \cos \left (\frac {7 t}{10}\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.887 |
|
|
\[ {}x^{\prime \prime }+\frac {x^{\prime }}{10}+x = 3 \cos \left (2 t \right ) \] \(r = -{\frac {399}{400}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.029 |
|
|
\[ {}x^{\prime \prime }-3 x^{\prime }+4 x = 0 \] \(r = -{\frac {7}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.845 |
|
|
\[ {}x^{\prime \prime }+6 x^{\prime }+9 x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x^{\prime \prime }+16 x = t \sin \left (t \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.638 |
|
|
\[ {}x^{\prime \prime }+x = {\mathrm e}^{t} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.738 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (x \right )+2 \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.612 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime } = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.053 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.675 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 2 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.315 |
|
|
\[ {}y^{\prime \prime } \left (2+x \right )^{5} = 1 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.001 |
|
|
\[ {}y^{\prime \prime } = 2 x \ln \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
0.646 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime } \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.02 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.675 |
|
|
\[ {}x y^{\prime \prime } = \left (2 x^{2}+1\right ) y^{\prime } \] \(r = \frac {4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.651 |
|
|
\[ {}x y^{\prime \prime } = y^{\prime }+x^{2} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.184 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+2 = 0 \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.143 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.529 |
|
|
\[ {}3 y^{\prime \prime }-2 y^{\prime }-8 y = 0 \] \(r = {\frac {25}{9}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.217 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.396 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-2 y = 0 \] \(r = 3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.228 |
|
|
\[ {}4 y^{\prime \prime }-8 y^{\prime }+5 y = 0 \] \(r = -{\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.309 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.461 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+3 y = 0 \] \(r = -2\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.696 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 3 \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.917 |
|
|
\[ {}y^{\prime \prime }-7 y^{\prime } = \left (-1+x \right )^{2} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.098 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = {\mathrm e}^{x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.07 |
|
|
\[ {}y^{\prime \prime }+7 y^{\prime } = {\mathrm e}^{-7 x} \] \(r = {\frac {49}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.112 |
|
|
\[ {}y^{\prime \prime }-8 y^{\prime }+16 y = \left (1-x \right ) {\mathrm e}^{4 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.448 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = {\mathrm e}^{5 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.397 |
|
|
\[ {}4 y^{\prime \prime }-3 y^{\prime } = x \,{\mathrm e}^{\frac {3 x}{4}} \] \(r = {\frac {9}{64}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.201 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = x \,{\mathrm e}^{4 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime \prime }+25 y = \cos \left (5 x \right ) \] \(r = -25\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.498 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )-\cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.541 |
|
|
\[ {}y^{\prime \prime }+16 y = \sin \left (4 x +\alpha \right ) \] \(r = -16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.692 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+8 y = {\mathrm e}^{2 x} \left (\sin \left (2 x \right )+\cos \left (2 x \right )\right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.261 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+8 y = {\mathrm e}^{2 x} \left (\sin \left (2 x \right )-\cos \left (2 x \right )\right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.008 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+13 y = {\mathrm e}^{-3 x} \cos \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.918 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = k \sin \left (k x +\alpha \right ) \] \(r = -k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.767 |
|
|
\[ {}y^{\prime \prime }+k^{2} y = k \] \(r = -k^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.478 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = -2 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.548 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = -2 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.509 |
|
|
\[ {}y^{\prime \prime }+9 y = 9 \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.54 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = x^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }+8 y^{\prime } = 8 x \] \(r = 16\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.697 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }+k^{2} y = {\mathrm e}^{x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.674 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 8 \,{\mathrm e}^{-2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 9 \,{\mathrm e}^{-3 x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.578 |
|
|
\[ {}7 y^{\prime \prime }-y^{\prime } = 14 x \] \(r = {\frac {1}{196}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.685 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime } = 3 x \,{\mathrm e}^{-3 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.924 |
|
|
\[ {}y^{\prime \prime }+5 y^{\prime }+6 y = 10 \left (1-x \right ) {\mathrm e}^{-2 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.618 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = 1+x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.779 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = \left (x^{2}+x \right ) {\mathrm e}^{x} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.029 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }-2 y = 8 \sin \left (2 x \right ) \] \(r = 6\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.922 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.885 |
|
|
\[ {}y^{\prime \prime }-2 m y^{\prime }+m^{2} y = \sin \left (n x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = {\mathrm e}^{-x} \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.977 |
|
|
\[ {}y^{\prime \prime }+a^{2} y = 2 \cos \left (m x \right )+3 \sin \left (m x \right ) \] \(r = -a^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.224 |
|
|
\[ {}y^{\prime \prime }-y^{\prime } = {\mathrm e}^{x} \sin \left (x \right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.687 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 4 \left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.516 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = 10 \,{\mathrm e}^{-2 x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}4 y^{\prime \prime }+8 y^{\prime } = x \sin \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.262 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.526 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-2 y = x^{2} {\mathrm e}^{4 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.611 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \left (x^{2}+x \right ) {\mathrm e}^{3 x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = x^{3} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.653 |
|
|
\[ {}y^{\prime \prime }+y = x^{2} \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.147 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.021 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = {\mathrm e}^{2 x} \left (\sin \left (x \right )+2 \cos \left (x \right )\right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.991 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = {\mathrm e}^{x}+{\mathrm e}^{-2 x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.628 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime } = x +{\mathrm e}^{-4 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.099 |
|
|
\[ {}y^{\prime \prime }-y = x +\sin \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.757 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = \left (\sin \left (x \right )+1\right ) {\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.906 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (2 x \right ) \sin \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.778 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime } = 2 \cos \left (4 x \right )^{2} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
5.691 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 4 x -2 \,{\mathrm e}^{x} \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 18 x -10 \cos \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.339 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 2+{\mathrm e}^{x} \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.815 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \left (5 x +4\right ) {\mathrm e}^{x}+{\mathrm e}^{-x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.892 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \,{\mathrm e}^{-x}+17 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.145 |
|
|
\[ {}2 y^{\prime \prime }-3 y^{\prime }-2 y = 5 \,{\mathrm e}^{x} \cosh \left (x \right ) \] \(r = {\frac {25}{16}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.052 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.514 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \cos \left (x \right )^{2}+{\mathrm e}^{x}+x^{2} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
10.406 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 10 \sin \left (x \right )+17 \sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.356 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}-{\mathrm e}^{-x}+{\mathrm e}^{x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.491 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-3 y = 2 x +{\mathrm e}^{-x}-2 \,{\mathrm e}^{3 x} \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.839 |
|
|
\[ {}y^{\prime \prime }+4 y = {\mathrm e}^{x}+4 \sin \left (2 x \right )+2 \cos \left (x \right )^{2}-1 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.427 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 6 x \,{\mathrm e}^{-x} \left (1-{\mathrm e}^{-x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.811 |
|
|
\[ {}y^{\prime \prime }+y = \cos \left (2 x \right )^{2}+\sin \left (\frac {x}{2}\right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.928 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 1+8 \cos \left (x \right )+{\mathrm e}^{2 x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sin \left (\frac {x}{2}\right )^{2} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.671 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 1+{\mathrm e}^{x}+\cos \left (x \right )+\sin \left (x \right ) \] \(r = {\frac {9}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.404 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = {\mathrm e}^{x} \left (1-2 \sin \left (x \right )^{2}\right )+10 x +1 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.007 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 4 x +\sin \left (x \right )+\sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.831 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = 1+2 \cos \left (x \right )+\cos \left (2 x \right )-\sin \left (2 x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.813 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y+1 = \sin \left (x \right )+x +x^{2} \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.04 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 18 \,{\mathrm e}^{-3 x}+8 \sin \left (x \right )+6 \cos \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.781 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+1 = 3 \sin \left (2 x \right )+\cos \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.422 |
|
|
\[ {}y^{\prime \prime }+y = 2 \sin \left (2 x \right ) \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.769 |
|
|
\[ {}y^{\prime \prime }+y = 2-2 x \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.507 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 9 x^{2}-12 x +2 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.638 |
|
|
\[ {}y^{\prime \prime }+9 y = 36 \,{\mathrm e}^{3 x} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.662 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 2 \,{\mathrm e}^{2 x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.657 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \left (12 x -7\right ) {\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.584 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{-x} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+9 y = 10 \sin \left (x \right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.787 |
|
|
\[ {}y^{\prime \prime }+y = 2 \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.646 |
|
|
\[ {}y^{\prime \prime }+4 y = \sin \left (x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.77 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.681 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = 2 x^{2} {\mathrm e}^{x} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 16 \,{\mathrm e}^{-x}+9 x -6 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.668 |
|
|
|
|||||
|
\[ {}y^{\prime \prime }-y^{\prime } = -5 \,{\mathrm e}^{-x} \left (\cos \left (x \right )+\sin \left (x \right )\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.992 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 4 \,{\mathrm e}^{x} \cos \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+5 y = \sin \left (x \right ) \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.538 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+5 y = 4 \cos \left (2 x \right )+\sin \left (2 x \right ) \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.633 |
|
|
\[ {}y^{\prime \prime }-y = 1 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.75 |
|
|
\[ {}y^{\prime \prime }-y = -2 \cos \left (x \right ) \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.386 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 4 \,{\mathrm e}^{-x} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.413 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 8 \,{\mathrm e}^{x}+9 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.382 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-5 y = 1 \] \(r = {\frac {21}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
0.471 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.661 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 2 \,{\mathrm e}^{-2 x} \left (9 \sin \left (2 x \right )+4 \cos \left (2 x \right )\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.591 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = {\mathrm e}^{-x} \left (9 x^{2}+5 x -12\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
0.493 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0 \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.964 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.899 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+6 y = 0 \] \(r = -\frac {6}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.697 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.774 |
|
|
\[ {}\left (2+x \right )^{2} y^{\prime \prime }+3 \left (2+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {15}{4 \left (2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.79 |
|
|
\[ {}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }+4 y = 0 \] \(r = -\frac {1}{\left (2 x +1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = x \left (6-\ln \left (x \right )\right ) \] \(r = -\frac {5}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.521 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = \sin \left (\ln \left (x \right )\right ) \] \(r = \frac {2}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.589 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = -\frac {16 \ln \left (x \right )}{x} \] \(r = \frac {15}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.35 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }-2 y = x^{2}-2 x +2 \] \(r = \frac {4}{x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.255 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{m} \] \(r = \frac {3}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.67 |
|
|
\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = 2 \ln \left (x \right )^{2}+12 x \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
3.657 |
|
|
\[ {}\left (1+x \right )^{3} y^{\prime \prime }+3 \left (1+x \right )^{2} y^{\prime }+\left (1+x \right ) y = 6 \ln \left (1+x \right ) \] \(r = -\frac {1}{4 \left (1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.355 |
|
|
\[ {}\left (-2+x \right )^{2} y^{\prime \prime }-3 \left (-2+x \right ) y^{\prime }+4 y = x \] \(r = -\frac {1}{4 \left (-2+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.848 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }+\left (-2+4 x \right ) y^{\prime }-8 y = 0 \] \(r = \frac {4 x^{2}+12 x +13}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.714 |
|
|
\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y = 0 \] \(r = \frac {8 x^{2}-8 x +3}{4 \left (x^{2}-x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[_Jacobi] |
✓ |
✓ |
1.088 |
|
|
\[ {}\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (1+x \right ) y^{\prime }+6 y = 6 \] \(r = \frac {3 x^{2}+12 x +18}{\left (2 x^{2}+3 x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.181 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sin \left (x \right )} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.5 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {1}{1+{\mathrm e}^{x}} \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.326 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\cos \left (x \right )^{3}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.586 |
|
|
\[ {}y^{\prime \prime }+y = \frac {1}{\sqrt {\sin \left (x \right )^{5} \cos \left (x \right )}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.762 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = \frac {{\mathrm e}^{x}}{x^{2}+1} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.472 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+2 y = \frac {{\mathrm e}^{-x}}{\sin \left (x \right )} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.582 |
|
|
\[ {}y^{\prime \prime }+y = \frac {2}{\sin \left (x \right )^{3}} \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{2 x} \cos \left ({\mathrm e}^{x}\right ) \] \(r = {\frac {1}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.529 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x^{2}+1\right ) y^{\prime } = 4 x^{3} {\mathrm e}^{x^{2}} \] \(r = \frac {4 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.488 |
|
|
\[ {}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime } = 1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.468 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x -1\right ) y^{\prime } = -4 x^{2} \] \(r = \frac {4 x^{2}-4 x +3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.764 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 1 \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.32 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = \frac {6+x}{x^{2}} \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
2.353 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime } = \frac {1}{x^{2}+1} \] \(r = \frac {1}{\left (x^{2}+1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.061 |
|
|
\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] \(r = \frac {x^{2}-4 x +6}{4 \left (-1+x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.839 |
|
|
\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-y = 4 \,{\mathrm e}^{x} \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
N/A |
8.398 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = 2 x -2 \] \(r = \frac {x^{4}-8 x^{3}+24 x^{2}-24 x +12}{4 \left (x^{2}-2 x \right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.733 |
|
|
\[ {}x^{\prime \prime }+x^{\prime }+x = 0 \] \(r = -{\frac {3}{4}}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.384 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+6 x = 0 \] \(r = -5\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.363 |
|
|
\[ {}x^{\prime \prime }+2 x^{\prime }+x = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.253 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] \(r = -\lambda \) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.363 |
|
|
\[ {}y^{\prime \prime }+\lambda y = 0 \] \(r = -\lambda \) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.679 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.101 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✗ |
N/A |
1.0 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.275 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] \(r = 1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.601 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.421 |
|
|
\[ {}y^{\prime \prime }+\alpha y^{\prime } = 0 \] \(r = \frac {\alpha ^{2}}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.14 |
|
|
\[ {}y^{\prime \prime }+\alpha ^{2} y = 1 \] \(r = -\alpha ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
111.799 |
|
|
\[ {}y^{\prime \prime }+y = 1 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.641 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] \(r = -\lambda ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.908 |
|
|
\[ {}y^{\prime \prime }+\lambda ^{2} y = 0 \] \(r = -\lambda ^{2}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.921 |
|
|
\[ {}x y^{\prime \prime }+y^{\prime } = 0 \] \(r = -\frac {1}{4 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.819 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.802 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+\frac {y}{4} = 0 \] \(r = \frac {-3-4 x}{16 x^{2}}\) \(L = [2]\) case used \(2\) poles order = \([2]\) \( O(\infty ) = 1\) |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.894 |
|
|
\[ {}y^{\prime \prime }+4 y = \cos \left (x \right )^{2} \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.67 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \pi ^{2}-x^{2} \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.481 |
|
|
\[ {}y^{\prime \prime }-4 y = \cos \left (\pi x \right ) \] \(r = 4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.514 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = \arcsin \left (\sin \left (x \right )\right ) \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.8 |
|
|
\[ {}y^{\prime \prime }+9 y = \sin \left (x \right )^{3} \] \(r = -9\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.135 |
|
|
|
|||||
|
|
|||||
|
|