|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.070 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.033 |
|
|
\[
{}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.744 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.338 |
|
|
\[
{}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.563 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.430 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.217 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.897 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.752 |
|
|
\[
{}\left (x y^{\prime \prime \prime }-y^{\prime \prime }\right )^{2} = {y^{\prime \prime \prime }}^{2}+1
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✓ |
1.102 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime } = x
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.190 |
|
|
\[
{}y^{\prime \prime } = x \,{\mathrm e}^{x}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.820 |
|
|
\[
{}\left (y^{\prime }-x y^{\prime \prime }\right )^{2} = 1+{y^{\prime \prime }}^{2}
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.876 |
|
|
\[
{}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{2} y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.563 |
|
|
\[
{}y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.143 |
|
|
\[
{}2 y^{\prime \prime } = {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.614 |
|
|
\[
{}y^{\prime \prime } y+2 y^{\prime }-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.349 |
|
|
\[
{}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.039 |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.041 |
|
|
\[
{}\left (x +2\right )^{2} y^{\prime \prime \prime }+\left (x +2\right ) y^{\prime \prime }+y^{\prime } = 1
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.546 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.538 |
|
|
\[
{}\left (-1+x \right )^{2} y^{\prime \prime }+4 \left (-1+x \right ) y^{\prime }+2 y = \cos \left (x \right )
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.855 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.316 |
|
|
\[
{}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
0.047 |
|
|
\[
{}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.203 |
|
|
\[
{}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
139.868 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.039 |
|
|
\[
{}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }+2 \tan \left (x \right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.233 |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.125 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.134 |
|
|
\[
{}y^{\prime \prime } y-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-y^{2} x^{2}
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
0.133 |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.849 |
|
|
\[
{}y^{\prime \prime } = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.442 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.328 |
|
|
\[
{}y^{\prime \prime }+y y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.014 |
|
|
\[
{}\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 x y^{\prime }+6 y = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.175 |
|
|
\[
{}\left (x^{2}-x \right ) y^{\prime \prime }+\left (4 x +2\right ) y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.411 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.518 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.564 |
|
|
\[
{}x \left (x +2 y\right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2} = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.253 |
|
|
\[
{}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.714 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-\frac {y^{\prime }}{x}+x^{2} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.497 |
|
|
\[
{}4 x^{2} y^{\prime \prime \prime }+8 x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.217 |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 \sin \left (x \right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.625 |
|
|
\[
{}\left [\begin {array}{c} 3 x^{\prime }+3 x+2 y={\mathrm e}^{t} \\ 4 x-3 y^{\prime }+3 y=3 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.515 |
|
|
\[
{}x^{\prime } = \frac {2 x}{t}
\] |
[_separable] |
✓ |
2.158 |
|
|
\[
{}x^{\prime } = -\frac {t}{x}
\] |
[_separable] |
✓ |
4.022 |
|
|
\[
{}x^{\prime } = -x^{2}
\] |
[_quadrature] |
✓ |
1.942 |
|
|
\[
{}x^{\prime \prime }+2 x^{\prime }+2 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.973 |
|
|
\[
{}x^{\prime } = {\mathrm e}^{-x}
\] |
[_quadrature] |
✓ |
1.588 |
|
|
\[
{}x^{\prime }+2 x = t^{2}+4 t +7
\] |
[[_linear, ‘class A‘]] |
✓ |
1.354 |
|
|
\[
{}2 t x^{\prime } = x
\] |
[_separable] |
✓ |
2.228 |
|
|
\[
{}t^{2} x^{\prime \prime }-6 x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.432 |
|
|
\[
{}2 x^{\prime \prime }-5 x^{\prime }-3 x = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.860 |
|
|
\[
{}x^{\prime } = x \left (1-\frac {x}{4}\right )
\] |
[_quadrature] |
✓ |
2.294 |
|
|
\[
{}x^{\prime } = x^{2}+t^{2}
\] |
[[_Riccati, _special]] |
✓ |
1.129 |
|
|
\[
{}x^{\prime } = t \cos \left (t^{2}\right )
\] |
[_quadrature] |
✓ |
0.904 |
|
|
\[
{}x^{\prime } = \frac {t +1}{\sqrt {t}}
\] |
[_quadrature] |
✓ |
0.730 |
|
|
\[
{}x^{\prime \prime } = -3 \sqrt {t}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.741 |
|
|
\[
{}x^{\prime } = t \,{\mathrm e}^{-2 t}
\] |
[_quadrature] |
✓ |
0.552 |
|
|
\[
{}x^{\prime } = \frac {1}{t \ln \left (t \right )}
\] |
[_quadrature] |
✓ |
0.466 |
|
|
\[
{}\sqrt {t}\, x^{\prime } = \cos \left (\sqrt {t}\right )
\] |
[_quadrature] |
✓ |
0.621 |
|
|
\[
{}x^{\prime } = \frac {{\mathrm e}^{-t}}{\sqrt {t}}
\] |
[_quadrature] |
✓ |
0.623 |
|
|
\[
{}x^{\prime }+t x^{\prime \prime } = 1
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.364 |
|
|
\[
{}x^{\prime } = \sqrt {x}
\] |
[_quadrature] |
✓ |
1.595 |
|
|
\[
{}x^{\prime } = {\mathrm e}^{-2 x}
\] |
[_quadrature] |
✓ |
3.519 |
|
|
\[
{}y^{\prime } = 1+y^{2}
\] |
[_quadrature] |
✓ |
3.349 |
|
|
\[
{}u^{\prime } = \frac {1}{5-2 u}
\] |
[_quadrature] |
✓ |
1.869 |
|
|
\[
{}x^{\prime } = a x+b
\] |
[_quadrature] |
✓ |
0.896 |
|
|
\[
{}Q^{\prime } = \frac {Q}{4+Q^{2}}
\] |
[_quadrature] |
✓ |
1.496 |
|
|
\[
{}x^{\prime } = {\mathrm e}^{x^{2}}
\] |
[_quadrature] |
✓ |
2.079 |
|
|
\[
{}y^{\prime } = r \left (a -y\right )
\] |
[_quadrature] |
✓ |
0.832 |
|
|
\[
{}x^{\prime } = \frac {2 x}{t +1}
\] |
[_separable] |
✓ |
1.956 |
|
|
\[
{}\theta ^{\prime } = t \sqrt {t^{2}+1}\, \sec \left (\theta \right )
\] |
[_separable] |
✓ |
2.038 |
|
|
\[
{}\left (2 u+1\right ) u^{\prime }-t -1 = 0
\] |
[_separable] |
✓ |
2.720 |
|
|
\[
{}R^{\prime } = \left (t +1\right ) \left (1+R^{2}\right )
\] |
[_separable] |
✓ |
2.274 |
|
|
\[
{}y^{\prime }+y+\frac {1}{y} = 0
\] |
[_quadrature] |
✓ |
8.546 |
|
|
\[
{}\left (t +1\right ) x^{\prime }+x^{2} = 0
\] |
[_separable] |
✓ |
1.516 |
|
|
\[
{}y^{\prime } = \frac {1}{2 y+1}
\] |
[_quadrature] |
✓ |
2.181 |
|
|
\[
{}x^{\prime } = \left (4 t -x\right )^{2}
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
2.411 |
|
|
\[
{}x^{\prime } = 2 t x^{2}
\] |
[_separable] |
✓ |
2.328 |
|
|
\[
{}x^{\prime } = t^{2} {\mathrm e}^{-x}
\] |
[_separable] |
✓ |
2.400 |
|
|
\[
{}x^{\prime } = x \left (4+x\right )
\] |
[_quadrature] |
✓ |
2.745 |
|
|
\[
{}x^{\prime } = {\mathrm e}^{t +x}
\] |
[_separable] |
✓ |
4.296 |
|
|
\[
{}T^{\prime } = 2 a t \left (T^{2}-a^{2}\right )
\] |
[_separable] |
✓ |
2.028 |
|
|
\[
{}y^{\prime } = t^{2} \tan \left (y\right )
\] |
[_separable] |
✓ |
1.996 |
|
|
\[
{}x^{\prime } = \frac {\left (4+2 t \right ) x}{\ln \left (x\right )}
\] |
[_separable] |
✓ |
3.212 |
|
|
\[
{}y^{\prime } = \frac {2 t y^{2}}{t^{2}+1}
\] |
[_separable] |
✓ |
2.402 |
|
|
\[
{}x^{\prime } = \frac {t^{2}}{1-x^{2}}
\] |
[_separable] |
✓ |
3.952 |
|
|
\[
{}x^{\prime } = 6 t \left (x-1\right )^{{2}/{3}}
\] |
[_separable] |
✓ |
3.656 |
|
|
\[
{}x^{\prime } = \frac {4 t^{2}+3 x^{2}}{2 t x}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.102 |
|
|
\[
{}x^{\prime } {\mathrm e}^{2 t}+2 x \,{\mathrm e}^{2 t} = {\mathrm e}^{-t}
\] |
[[_linear, ‘class A‘]] |
✓ |
1.866 |
|
|
\[
{}\frac {x^{\prime }+t x^{\prime \prime }}{t} = -2
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.845 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 t y}{t^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.972 |
|
|
\[
{}y^{\prime } = -y^{2} {\mathrm e}^{-t^{2}}
\] |
[_separable] |
✓ |
2.312 |
|
|
\[
{}x^{\prime } = 2 t^{3} x-6
\] |
[_linear] |
✓ |
1.504 |
|
|
\[
{}\cos \left (t \right ) x^{\prime }-2 x \sin \left (x\right ) = 0
\] |
[_separable] |
✓ |
2.726 |
|
|
\[
{}x^{\prime } = t -x^{2}
\] |
[[_Riccati, _special]] |
✓ |
1.047 |
|
|
\[
{}7 t^{2} x^{\prime } = 3 x-2 t
\] |
[_linear] |
✓ |
1.315 |
|
|
\[
{}x x^{\prime } = 1-t x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
0.757 |
|