|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+4 y = \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.951 |
|
|
\[
{}y^{\prime \prime }+5 y = \cos \left (\sqrt {5}\, x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.383 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{x}+{\mathrm e}^{-x}+\sin \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.635 |
|
|
\[
{}y^{\prime \prime }-y = x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.049 |
|
|
\[
{}y^{\prime \prime }+2 y = x^{3}+x^{2}+{\mathrm e}^{-2 x}+\cos \left (3 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
12.392 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-y = {\mathrm e}^{x} \cos \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.697 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = \frac {{\mathrm e}^{2 x}}{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.060 |
|
|
\[
{}y^{\prime \prime }-y = x \,{\mathrm e}^{3 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.016 |
|
|
\[
{}y^{\prime \prime }+5 y^{\prime }+6 y = {\mathrm e}^{-2 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.121 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y = x +x^{2} \ln \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.632 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y = \ln \left (x \right )^{2}-\ln \left (x^{2}\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.696 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime } = x +\sin \left (\ln \left (x \right )\right )
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.364 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+y^{\prime } x -y = 3 x^{4}
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.244 |
|
|
\[
{}\left (x +1\right )^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }-y = \ln \left (x +1\right )^{2}+x -1
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.253 |
|
|
\[
{}\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y = 6 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.201 |
|
|
\[
{}x y^{\prime \prime }-\left (x +2\right ) y^{\prime }+2 y = 0
\] |
[_Laguerre] |
✓ |
0.954 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y = 2
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.701 |
|
|
\[
{}\left (x^{2}+4\right ) y^{\prime \prime }-2 y^{\prime } x +2 y = 8
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.778 |
|
|
\[
{}\left (x +1\right ) y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+\left (x +2\right ) y = \left (x^{2}+2 x +1\right ) {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.236 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-10 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.707 |
|
|
\[
{}x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+\left (x^{2}+3 x +3\right ) y = \left (-x^{2}+6\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.988 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.933 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = \left (x^{2}-x +1\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.452 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.288 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+y = \frac {x +1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
107.497 |
|
|
\[
{}x^{8} y^{\prime \prime }+4 x^{7} y^{\prime }+y = \frac {1}{x^{3}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.540 |
|
|
\[
{}\left (x \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+y \cos \left (x \right ) = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.055 |
|
|
\[
{}x y^{\prime \prime }-3 y^{\prime }+\frac {3 y}{x} = x +2
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.563 |
|
|
\[
{}\left (x +1\right ) y^{\prime \prime }-\left (3 x +4\right ) y^{\prime }+3 y = \left (3 x +2\right ) {\mathrm e}^{3 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.383 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (9 x^{2}+6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.248 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+4 y x = 4
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.837 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y = \frac {-x^{2}+1}{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.851 |
|
|
\[
{}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.350 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 y^{\prime } x = \frac {2}{x^{3}}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.314 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } = -\frac {2}{x}-\ln \left (x \right )
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.154 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime \prime } = x^{2}
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.101 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{3} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.492 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.401 |
|
|
\[
{}y y^{\prime \prime } = {y^{\prime }}^{2} \left (1-y^{\prime } \cos \left (y\right )+y y^{\prime } \sin \left (y\right )\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
1.115 |
|
|
\[
{}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 y^{\prime } x -4 y = 8
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.049 |
|
|
\[
{}\left (2 x^{3}-1\right ) y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+6 y^{\prime } x = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.329 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
4.335 |
|
|
\[
{}\left (x +2 y\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+2 y^{\prime } = 2
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.581 |
|
|
\[
{}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x
\] |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
0.054 |
|
|
\[
{}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
0.057 |
|
|
\[
{}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
0.056 |
|
|
\[
{}2 \left (1+y\right ) y^{\prime \prime }+2 {y^{\prime }}^{2}+y^{2}+2 y = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
4.394 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }-y^{\prime }+y=-{\mathrm e}^{t} \\ x+y^{\prime }-y={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.473 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=t \\ 5 x+y^{\prime }+3 y=t^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.714 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+x+2 y^{\prime }+7 y={\mathrm e}^{t}+2 \\ -2 x+y^{\prime }+3 y={\mathrm e}^{t}-1 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.717 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+3 y={\mathrm e}^{-t}-1 \\ x^{\prime }+2 x+y^{\prime }+3 y={\mathrm e}^{2 t}+1 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.194 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }-x+y^{\prime }+2 y=1+{\mathrm e}^{t} \\ y^{\prime }+2 y+z^{\prime }+z={\mathrm e}^{t}+2 \\ x^{\prime }-x+z^{\prime }+z=3+{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.551 |
|
|
\[
{}\left (1-x \right ) y^{\prime } = x^{2}-y
\] |
[_linear] |
✓ |
0.470 |
|
|
\[
{}y^{\prime } x = 1-x +2 y
\] |
[_linear] |
✓ |
0.508 |
|
|
\[
{}y^{\prime } x = 1-x +2 y
\] |
[_linear] |
✓ |
1.225 |
|
|
\[
{}y^{\prime } = 2 x^{2}+3 y
\] |
[[_linear, ‘class A‘]] |
✓ |
0.485 |
|
|
\[
{}\left (x +1\right ) y^{\prime } = x^{2}-2 x +y
\] |
[_linear] |
✓ |
0.466 |
|
|
\[
{}y^{\prime \prime }+y x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.439 |
|
|
\[
{}y^{\prime \prime }+2 x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.441 |
|
|
\[
{}y^{\prime \prime }-y^{\prime } x +x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.483 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +p \left (p +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
0.691 |
|
|
\[
{}y^{\prime \prime }+x^{2} y = x^{2}+x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.471 |
|
|
\[
{}2 \left (x^{3}+x^{2}\right ) y^{\prime \prime }-\left (-3 x^{2}+x \right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.813 |
|
|
\[
{}4 x y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.807 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.766 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y x = 0
\] |
[_Lienard] |
✓ |
0.628 |
|
|
\[
{}x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.717 |
|
|
\[
{}x y^{\prime \prime }-2 y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.177 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+y x = 0
\] |
[_Lienard] |
✓ |
0.724 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.904 |
|
|
\[
{}2 x y^{\prime \prime }+y^{\prime }-y = x +1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.948 |
|
|
\[
{}2 x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.808 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.757 |
|
|
\[
{}z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.562 |
|
|
\[
{}x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.978 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }-\left (2 x +1\right ) \left (-y+y^{\prime } x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.980 |
|
|
\[
{}x^{3} \left (x +1\right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.003 |
|
|
\[
{}x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.006 |
|
|
\[
{}2 \left (2-x \right ) x^{2} y^{\prime \prime }-\left (4-x \right ) x y^{\prime }+\left (3-x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.997 |
|
|
\[
{}\left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.327 |
|
|
\[
{}x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.666 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 \left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.962 |
|
|
\[
{}x y^{\prime \prime }+\left (x^{3}+1\right ) y^{\prime }+b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.721 |
|
|
\[
{}\left (x -1\right ) \left (x -2\right ) y^{\prime \prime }+\left (4 x -6\right ) y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.622 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.485 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime } x +8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.488 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +12 y = 0
\] |
[_Gegenbauer] |
✓ |
0.584 |
|
|
\[
{}y^{\prime \prime } = \left (x -1\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.468 |
|
|
\[
{}x \left (x +2\right ) y^{\prime \prime }+2 \left (x +1\right ) y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.784 |
|
|
\[
{}x y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.092 |
|
|
\[
{}y^{\prime \prime }+\left ({\mathrm e}^{x}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.654 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }-3 y^{\prime } x -y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.193 |
|
|
\[
{}2 x y^{\prime \prime }-y^{\prime }+x^{2} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.702 |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }-y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.013 |
|
|
\[
{}y^{\prime \prime }-x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.447 |
|
|
\[
{}x \left (x +2\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.850 |
|
|
\[
{}x y^{\prime \prime }+\left (\frac {1}{2}-x \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.797 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.769 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {9}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.709 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}+\frac {25}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.719 |
|