Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] \(r = \frac {3}{\left (x^{2}-1\right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -12\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 10\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
160.967 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y = 0 \] \(r = \frac {x^{4} \left (x^{6}-14\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -10\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.786 |
|
|
\[ {}\left (4 x +2\right ) y^{\prime \prime }-4 y^{\prime }-\left (6+4 x \right ) y = 0 \] \(r = \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.907 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (x +5\right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-14 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2, 4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([6]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.5 |
|
|
\[ {}x y^{\prime \prime }+\left (-6+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}+48}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.881 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.709 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2, 4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.475 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.777 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([6]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.983 |
|
|
\[ {}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y = 0 \] \(r = \frac {-x +36}{4 x \left (-1+x \right )^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -12\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 10\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
168.78 |
|
|
\[ {}y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y = 0 \] \(r = \frac {x^{4} \left (x^{6}-14\right )}{4}\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -10\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.736 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (x +5\right ) y^{\prime }-\left (2-3 x \right ) y = 0 \] \(r = \frac {x^{2}-14 x +21}{16 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2, 4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([6]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.621 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (3 x -2\right ) y = 0 \] \(r = -\frac {2}{9 x^{2}}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.447 |
|
|
\[ {}x \left (-1+x \right )^{2} y^{\prime \prime }-2 y = 0 \] \(r = \frac {2}{\left (-1+x \right )^{2} x}\) \(L = [1, 2, 4, 6, 12]\) case used \(1\) poles order = \([1, 2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.543 |
|
|
\[ {}x y^{\prime \prime }+\left (-6+x \right ) y^{\prime }-3 y = 0 \] \(r = \frac {x^{2}+48}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
0.532 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] \(r = -1\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[_Jacobi] |
✓ |
✓ |
0.75 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+4 x y = 0 \] \(r = -4\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 3\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.526 |
|
|
\[ {}u^{\prime \prime }+2 u^{\prime }+u = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2, 4]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.519 |
|
|
\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y = 0 \] \(r = \frac {-16 x^{4}+3}{4 x^{2}}\) \(L = [1, 2]\) case used \(1\) poles order = \([2]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.608 |
|
|
\[ {}y^{\prime \prime } = 0 \] \(r = 0\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([1, 2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 4\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([4]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 1\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.479 |
|
|
\[ {}y^{\prime \prime } = \left (x^{2}+3\right ) y \] \(r = x^{2}+3\) \(L = [1]\) case used \(1\) poles order = \([]\) \( O(\infty ) = -2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = infinity\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([]\) \( O(\infty ) = 0\) |
1 |
1 |
1 |
kovacic |
[[_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 = \([2]\) \( O(\infty ) = 2\) |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.688 |
|
|
|
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|
|
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|
|