These are second order ode’s where one solution is known and reduction of order is used to find the second solution. Number of problems in this table is 160
Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.385 |
|
|
\[ {}4 y^{\prime \prime }-4 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.24 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.628 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }-\left (2+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.624 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
0.609 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
0.595 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.702 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.454 |
|
|
\[ {}t^{2} y^{\prime \prime }+2 t y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.44 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.449 |
|
|
\[ {}t^{2} y^{\prime \prime }-t \left (2+t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.625 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.77 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.718 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.54 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.675 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = \left (2 x +1\right )^{2} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.781 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \frac {4}{x^{2}} \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.554 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.571 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \frac {1}{1+{\mathrm e}^{-x}} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.637 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = 7 x^{\frac {3}{2}} {\mathrm e}^{x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.574 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 4 \sqrt {x}\, {\mathrm e}^{x} \left (1+4 x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.956 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+2 y = {\mathrm e}^{x} \sec \left (x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.749 |
|
|
\[ {}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 8 \,{\mathrm e}^{-x \left (2+x \right )} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.844 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = -6 x -4 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.626 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x \left (-1+x \right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y = x^{3} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.883 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y = x^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.743 |
|
|
\[ {}\left (1-2 x \right ) y^{\prime \prime }+2 y^{\prime }+\left (2 x -3\right ) y = \left (4 x^{2}-4 x +1\right ) {\mathrm e}^{x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.822 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 4 x^{4} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.649 |
|
|
\[ {}2 x y^{\prime \prime }+\left (1+4 x \right ) y^{\prime }+\left (2 x +1\right ) y = 3 \sqrt {x}\, {\mathrm e}^{-x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.824 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = -{\mathrm e}^{-x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.8 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y = 4 x^{\frac {5}{2}} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.828 |
|
|
\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+8 y = 4 x^{2} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.666 |
|
|
\[ {}x y^{\prime \prime }+\left (2-2 x \right ) y^{\prime }+\left (-2+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.918 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.554 |
|
|
\[ {}x^{2} \ln \left (x \right )^{2} y^{\prime \prime }-2 x \ln \left (x \right ) y^{\prime }+\left (2+\ln \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.474 |
|
|
\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.699 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.877 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.526 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.685 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.882 |
|
|
\[ {}4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.846 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.726 |
|
|
\[ {}\left (2 x +1\right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.085 |
|
|
\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.777 |
|
|
\[ {}x y^{\prime \prime }-\left (1+4 x \right ) y^{\prime }+\left (4 x +2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.668 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 4 x^{4} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.959 |
|
|
\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.238 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y = \left (1+x \right )^{3} {\mathrm e}^{x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.324 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = x^{2} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
|
\[ {}\left (x^{2}-4\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 2+x \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.134 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.26 |
|
|
\[ {}x y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.375 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.418 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
0.384 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}+4 x^{2} y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.375 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.425 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.385 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+2 y = 8 x^{2} {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.377 |
|
|
\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = 8 x^{4} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.298 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = 15 \,{\mathrm e}^{3 x} \sqrt {x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.332 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 4 \,{\mathrm e}^{2 x} \ln \left (x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.333 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = \sqrt {x}\, \ln \left (x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.283 |
|
|
\[ {}x^{2} \left (2-x \right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.415 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.265 |
|
|
\[ {}x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.429 |
|
|
\[ {}3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (-2+3 x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.51 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.253 |
|
|
\[ {}x \left (1+x \right ) y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.448 |
|
|
\[ {}x^{2} y^{\prime \prime }-7 x y^{\prime }+15 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.408 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.408 |
|
|
\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.406 |
|
|
\[ {}x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[_Laguerre] |
✓ |
✓ |
0.429 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.453 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.325 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.373 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.414 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.345 |
|
|
\[ {}x y^{\prime \prime }+3 y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.642 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.668 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
1.02 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.14 |
|
|
\[ {}y^{\prime \prime }-\frac {x y^{\prime }}{-1+x}+\frac {y}{-1+x} = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.676 |
|
|
\[ {}x^{2} y^{\prime \prime }-x \left (2+x \right ) y^{\prime }+\left (2+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.072 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime \prime }-y = 3 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.342 |
|
|
\[ {}y^{\prime \prime }+y = -8 \sin \left (3 x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.286 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2}+2 x +2 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.068 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = \frac {-1+x}{x} \] |
1 |
1 |
1 |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
26.779 |
|
|
\[ {}x^{\prime \prime }+t x^{\prime }+x = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.423 |
|
|
\[ {}x^{\prime \prime }-t x^{\prime }+x = 0 \] |
1 |
1 |
1 |
[_Hermite] |
✓ |
✓ |
0.525 |
|
|
\[ {}x^{\prime \prime }-2 a x^{\prime }+a^{2} x = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.214 |
|
|
\[ {}x^{\prime \prime }-\frac {\left (2+t \right ) x^{\prime }}{t}+\frac {\left (2+t \right ) x}{t^{2}} = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.581 |
|
|
\[ {}t^{2} x^{\prime \prime }+t x^{\prime }+\left (t^{2}-\frac {1}{4}\right ) x = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.727 |
|
|
\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.471 |
|
|
\[ {}\left (1+x \right )^{2} y^{\prime \prime }-3 \left (1+x \right ) y^{\prime }+3 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.588 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[_Gegenbauer] |
✓ |
✓ |
0.613 |
|
|
\[ {}\left (x^{2}-x +1\right ) y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+\left (1+x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.006 |
|
|
\[ {}\left (2 x +1\right ) y^{\prime \prime }-4 \left (1+x \right ) y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.685 |
|
|
\[ {}\left (x^{3}-x^{2}\right ) y^{\prime \prime }-\left (x^{3}+2 x^{2}-2 x \right ) y^{\prime }+\left (2 x^{2}+2 x -2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.828 |
|
|
\[ {}t^{2} y^{\prime \prime }-\left (t^{2}+2 t \right ) y^{\prime }+\left (2+t \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.594 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x^{\prime \prime }-x^{\prime } t \sin \left (t \right )-x \sin \left (t \right ) = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.856 |
|
|
\[ {}\left (-t^{2}+t \right ) x^{\prime \prime }+\left (-t^{2}+2\right ) x^{\prime }+\left (2-t \right ) x = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[_Hermite] |
✓ |
✓ |
0.515 |
|
|
\[ {}\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.687 |
|
|
\[ {}\left (\tan \left (x \right )^{2}-1\right ) y^{\prime \prime }-4 \tan \left (x \right )^{3} y^{\prime }+2 y \sec \left (x \right )^{4} = \left (\tan \left (x \right )^{2}-1\right ) \left (1-2 \sin \left (x \right )^{2}\right ) \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.349 |
|
|
\[ {}x^{3} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.25 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.411 |
|
|
\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \] |
1 |
1 |
1 |
[_Lienard] |
✓ |
✓ |
0.493 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.22 |
|
|
\[ {}y^{\prime \prime }-10 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.215 |
|
|
\[ {}x^{2} y^{\prime \prime }-6 x y^{\prime }+12 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.449 |
|
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.421 |
|
|
\[ {}4 x^{2} y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.615 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.625 |
|
|
\[ {}y^{\prime \prime }-\frac {y^{\prime }}{x}-4 x^{2} y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.506 |
|
|
\[ {}y^{\prime \prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.251 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.52 |
|
|
\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1+\cos \left (x \right )^{2}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.813 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.589 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.575 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.577 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+3 y = 9 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.366 |
|
|
\[ {}y^{\prime \prime }-6 y^{\prime }+8 y = {\mathrm e}^{4 x} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.416 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = \sqrt {x} \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.429 |
|
|
\[ {}x^{2} y^{\prime \prime }-20 y = 27 x^{5} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.399 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y = 8 \,{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.468 |
|
|
\[ {}\left (1+x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1+x \right )^{2} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.527 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.283 |
|
|
\[ {}y^{\prime \prime }+6 y^{\prime }+8 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.286 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.441 |
|
|
\[ {}y^{\prime \prime }+10 y^{\prime }+25 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+9 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.563 |
|
|
\[ {}y^{\prime \prime }+49 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.544 |
|
|
\[ {}t^{2} y^{\prime \prime }+4 t y^{\prime }-4 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.521 |
|
|
\[ {}t^{2} y^{\prime \prime }+6 t y^{\prime }+6 y = 0 \] |
1 |
1 |
1 |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.578 |
|
|
\[ {}t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.758 |
|
|
\[ {}t^{2} y^{\prime \prime }+3 t y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
0.548 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (36 t^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.005 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+16 t y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.822 |
|
|
\[ {}y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.766 |
|
|
\[ {}y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y = 0 \] |
1 |
0 |
0 |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.922 |
|
|
\[ {}t^{2} y^{\prime \prime }-4 t y^{\prime }+\left (t^{2}+6\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.833 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
[_Lienard] |
✓ |
✓ |
0.618 |
|
|
\[ {}4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (16 t^{2}-1\right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.963 |
|
|
\[ {}\left (t +1\right )^{2} y^{\prime \prime }-2 \left (t +1\right ) y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.749 |
|
|
\[ {}t y^{\prime \prime }+2 y^{\prime }+t y = 0 \] |
1 |
1 |
1 |
[_Lienard] |
✓ |
✓ |
0.674 |
|
|
\[ {}x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.287 |
|
|
\[ {}y^{\prime \prime }+\left (\tan \left (x \right )-2 \cot \left (x \right )\right ) y^{\prime }+2 \cot \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.844 |
|
|
\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.562 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = 1 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.324 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }-3 y = 5 x^{4} \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.283 |
|
|
\[ {}\left (-1+x \right ) y^{\prime \prime }-x y^{\prime }+y = \left (-1+x \right )^{2} {\mathrm e}^{x} \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.352 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y \,{\mathrm e}^{-2 x} = {\mathrm e}^{-3 x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.536 |
|
|
\[ {}\left (x^{4}-x^{3}\right ) y^{\prime \prime }+\left (2 x^{3}-2 x^{2}-x \right ) y^{\prime }-y = \frac {\left (-1+x \right )^{2}}{x} \] |
1 |
1 |
1 |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.374 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+y \,{\mathrm e}^{2 x} = {\mathrm e}^{2 x} x -1 \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.623 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y = x^{2} \left (2 x -3\right ) \] |
1 |
1 |
1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.359 |
|
|
|
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|
|
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