|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime } = x y^{\prime }
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.761 |
|
|
\[
{}y^{\prime \prime } = \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.770 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{x}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.738 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.531 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-4 x y^{\prime \prime }+6 y^{\prime } = 4
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.176 |
|
|
\[
{}y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x \left (a^{2}-x^{2}\right )} = \frac {x^{2}}{a \left (a^{2}-x^{2}\right )}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.500 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a x = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.215 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime } = a x
\] |
[[_2nd_order, _missing_y]] |
✓ |
34.424 |
|
|
\[
{}x y^{\prime \prime }+x {y^{\prime }}^{2}-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.427 |
|
|
\[
{}x y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.282 |
|
|
\[
{}y^{\prime }-x y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2}}{a} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.893 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime } = x
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.921 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2}}{a} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.687 |
|
|
\[
{}y^{\prime \prime }+y y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.018 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2} = 1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
3.112 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.346 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.035 |
|
|
\[
{}y^{\prime \prime } = a {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.447 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
2.274 |
|
|
\[
{}y y^{\prime \prime }+\sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}} = {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
123.204 |
|
|
\[
{}a y^{\prime \prime } = y^{\prime }
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.126 |
|
|
\[
{}a^{2} y^{\prime \prime } y^{\prime } = x
\] |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
1.727 |
|
|
\[
{}y^{\prime \prime \prime } y^{\prime \prime } = 2
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✓ |
0.471 |
|
|
\[
{}y^{\prime \prime } = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.378 |
|
|
\[
{}a y^{\prime \prime } = \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.450 |
|
|
\[
{}y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.545 |
|
|
\[
{}a^{2} {y^{\prime \prime }}^{2} = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
11.145 |
|
|
\[
{}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.672 |
|
|
\[
{}y^{\prime } = x y^{\prime \prime }+\sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
6.696 |
|
|
\[
{}a^{2} y^{\prime \prime \prime \prime } = y^{\prime \prime }
\] |
[[_high_order, _missing_x]] |
✓ |
0.057 |
|
|
\[
{}y^{\prime \prime \prime \prime }+a^{2} y^{\prime \prime } = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.064 |
|
|
\[
{}y^{\left (5\right )}-n^{2} y^{\prime \prime \prime } = {\mathrm e}^{a x}
\] |
[[_high_order, _missing_y]] |
✓ |
0.116 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+a^{2} y^{\prime \prime } = 0
\] |
[[_high_order, _missing_y]] |
✓ |
0.566 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime } = \lambda y^{\prime \prime }
\] |
[[_high_order, _missing_y]] |
✓ |
0.468 |
|
|
\[
{}n \,x^{3} y^{\prime \prime \prime } = y-x y^{\prime }
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.117 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2} = 3 y y^{\prime }
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.974 |
|
|
\[
{}2 x^{2} y y^{\prime \prime }+y^{2} = x^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.144 |
|
|
\[
{}x^{2} y^{\prime \prime } = \sqrt {m \,x^{2} {y^{\prime }}^{3}+n y^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.319 |
|
|
\[
{}x^{4} y^{\prime \prime } = \left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.132 |
|
|
\[
{}x^{4} y^{\prime \prime }-x^{3} y^{\prime } = x^{2} {y^{\prime }}^{2}-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.139 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 y^{2}-6 y = x^{4} {y^{\prime }}^{2}
\] |
[NONE] |
✗ |
0.136 |
|
|
\[
{}y^{\prime \prime } = {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.798 |
|
|
\[
{}y^{\prime \prime }+a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.671 |
|
|
\[
{}a y^{\prime \prime \prime } = y^{\prime \prime }
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.048 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+1 = 0
\] |
[[_high_order, _quadrature]] |
✓ |
0.214 |
|
|
\[
{}y^{\prime \prime \prime } = \sin \left (x \right )^{2}
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.150 |
|
|
\[
{}y^{\prime \prime } = \frac {1}{\sqrt {a y}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
30.421 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.438 |
|
|
\[
{}-a y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.967 |
|
|
\[
{}\sin \left (y\right )^{3} y^{\prime \prime } = \cos \left (y\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
74.925 |
|
|
\[
{}{\mathrm e}^{x} \left (x y^{\prime \prime }-y^{\prime }\right ) = x^{3}
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.830 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.283 |
|
|
\[
{}2 x y^{\prime \prime \prime } y^{\prime \prime } = {y^{\prime \prime }}^{2}-a^{2}
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✓ |
0.823 |
|
|
\[
{}y y^{\prime \prime }+\sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}} = {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
110.907 |
|
|
\[
{}\left (x^{3}-4 x \right ) y^{\prime \prime \prime }+\left (9 x^{2}-4\right ) y^{\prime \prime }+18 x y^{\prime }+6 y = 6
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.275 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.684 |
|
|
\[
{}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0
\] |
[_Laguerre] |
✓ |
0.826 |
|
|
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime } = y+{\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.349 |
|
|
\[
{}\left (x +1\right ) y^{\prime \prime }-2 \left (x +3\right ) y^{\prime }+\left (x +5\right ) y = {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.995 |
|
|
\[
{}\left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.036 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }-y = X
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.826 |
|
|
\[
{}y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime }+x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.038 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
0.104 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+2 x \right ) y^{\prime }+\left (x +2\right ) y = x^{3} {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.201 |
|
|
\[
{}y^{\prime \prime }-a x y^{\prime }+a^{2} \left (-1+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.125 |
|
|
\[
{}\left (2 x^{3}-a \right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.091 |
|
|
\[
{}y^{\prime \prime }+4 x y^{\prime }+4 x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.809 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.204 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.225 |
|
|
\[
{}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
33.344 |
|
|
\[
{}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
50.863 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{3}+6 x^{2}+4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.905 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.924 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = \sec \left (x \right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
27.960 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.880 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.734 |
|
|
\[
{}y^{\prime \prime }+2 n \cot \left (n x \right ) y^{\prime }+\left (m^{2}-n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.679 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {y \left (-8+\sqrt {x}+x \right )}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.067 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 n x y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.484 |
|
|
\[
{}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.950 |
|
|
\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.393 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.099 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.323 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+m^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.530 |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\sin \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.495 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2}+\sin \left (x \right ) \cos \left (x \right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.083 |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.213 |
|
|
\[
{}y^{\prime \prime }+\left (\tan \left (x \right )-1\right )^{2} y^{\prime }-n \left (n -1\right ) y \sec \left (x \right )^{4} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.732 |
|
|
\[
{}y^{\prime \prime }+\left (3 \sin \left (x \right )-\cot \left (x \right )\right ) y^{\prime }+2 \sin \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
6.004 |
|
|
\[
{}3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.191 |
|
|
\[
{}x y^{\prime \prime }+\left (-2+x \right ) y^{\prime }-2 y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.033 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime }-\left (x^{2}+1\right ) y = {\mathrm e}^{-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.223 |
|
|
\[
{}\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.416 |
|
|
\[
{}y^{\prime \prime }+y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.768 |
|
|
\[
{}y^{\prime \prime }+y = \csc \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.783 |
|
|
\[
{}y^{\prime \prime }+4 y = 4 \tan \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
6.168 |
|
|
\[
{}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (1-x \right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.531 |
|
|
\[
{}y^{\prime \prime }-y = \frac {2}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.352 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }-\left (x^{2}+1\right ) y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.474 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = -4 x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.440 |
|