|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}r^{\prime \prime } = -\frac {k}{r^{2}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
70.770 |
|
|
\[
{}y^{\prime \prime } = \frac {3 k y^{2}}{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.089 |
|
|
\[
{}y^{\prime \prime } = 2 k y^{3}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.908 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.594 |
|
|
\[
{}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.288 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{3}-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.620 |
|
|
\[
{}y y^{\prime \prime }-3 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.349 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.734 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (1+y^{\prime }\right ) = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.428 |
|
|
\[
{}\left (1+y\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.637 |
|
|
\[
{}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.853 |
|
|
\[
{}y^{\prime \prime } = 2 y y^{\prime }
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.004 |
|
|
\[
{}2 y^{\prime \prime } = {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
18.148 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime } = 1
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.298 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } = x^{2}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.486 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+y y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.206 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.763 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (1+y\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.144 |
|
|
\[
{}-a y^{3}-\frac {b}{x^{{3}/{2}}}+y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
8.006 |
|
|
\[
{}a x y^{3}+b y^{2}+y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
2.117 |
|
|
\[
{}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0
\] |
[_Abel] |
✓ |
3.974 |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✗ |
30.313 |
|
|
\[
{}x^{2} y^{\prime }+x y^{3}+y^{2} a = 0
\] |
[_rational, _Abel] |
✗ |
0.882 |
|
|
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✗ |
1.863 |
|
|
\[
{}y^{\prime }+y \tan \left (x \right ) = 0
\] |
[_separable] |
✓ |
1.352 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.228 |
|
|
\[
{}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.924 |
|
|
\[
{}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+1 = 0
\] |
[_quadrature] |
✓ |
0.296 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{a x}+a y
\] |
[[_linear, ‘class A‘]] |
✓ |
0.686 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right )^{3} = a^{2} {y^{\prime \prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
5.758 |
|
|
\[
{}\left (x +1\right ) y+\left (1-y\right ) x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.303 |
|
|
\[
{}y^{\prime } = a y^{2} x
\] |
[_separable] |
✓ |
1.275 |
|
|
\[
{}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.576 |
|
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
2.901 |
|
|
\[
{}\frac {x}{1+y} = \frac {y y^{\prime }}{x +1}
\] |
[_separable] |
✓ |
1.452 |
|
|
\[
{}y^{\prime }+b^{2} y^{2} = a^{2}
\] |
[_quadrature] |
✓ |
1.534 |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
1.775 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right ) = \cos \left (x \right ) \sin \left (y\right ) y^{\prime }
\] |
[_separable] |
✓ |
2.620 |
|
|
\[
{}a x y^{\prime }+2 y = x y y^{\prime }
\] |
[_separable] |
✓ |
1.422 |
|
|
\[
{}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.935 |
|
|
\[
{}y^{\prime \prime }+x y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.463 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.931 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y = 2
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.885 |
|
|
\[
{}y^{\prime \prime }+a \,x^{2} y = x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.510 |
|
|
\[
{}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.175 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.935 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.897 |
|
|
\[
{}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.026 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.944 |
|
|
\[
{}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.936 |
|
|
\[
{}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.138 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.076 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+x y^{\prime }-n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.675 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.659 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.720 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+p x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.685 |
|
|
\[
{}x y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.157 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.116 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.283 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.225 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-3 x^{2}+1\right ) y^{\prime }-x y = 0
\] |
[[_elliptic, _class_I]] |
✓ |
0.775 |
|
|
\[
{}y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
0.171 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.273 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+x y = 0
\] |
[[_elliptic, _class_II]] |
✓ |
0.740 |
|
|
\[
{}4 x \left (1-x \right ) y^{\prime \prime }-4 y^{\prime }-y = 0
\] |
[_Jacobi] |
✓ |
1.337 |
|
|
\[
{}x^{3} y^{\prime \prime }+y = x^{{3}/{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.170 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = \sqrt {x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.143 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 3 x^{2}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.029 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0
\] |
[_Jacobi] |
✓ |
0.953 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.297 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0
\] |
[_Jacobi] |
✓ |
0.872 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (-2 x +1\right ) y^{\prime }}{3}+\frac {20 y}{9} = 0
\] |
[_Jacobi] |
✓ |
0.908 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+4 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.847 |
|
|
\[
{}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.638 |
|
|
\[
{}y^{2}+y^{\prime } = \frac {a^{2}}{x^{4}}
\] |
[_rational, _Riccati] |
✓ |
1.431 |
|
|
\[
{}u^{\prime \prime }-\frac {a^{2} u}{x^{{2}/{3}}} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.821 |
|
|
\[
{}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.247 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.427 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.442 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.102 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.774 |
|
|
\[
{}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.179 |
|
|
\[
{}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.966 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.439 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.083 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.125 |
|
|
\[
{}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.126 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{2 x} y = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.656 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{4 x} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.729 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.701 |
|
|
\[
{}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.226 |
|
|
\[
{}y^{\prime } = y
\] |
[_quadrature] |
✓ |
0.957 |
|
|
\[
{}x y^{\prime } = y
\] |
[_separable] |
✓ |
1.503 |
|
|
\[
{}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
3.141 |
|
|
\[
{}y^{\prime } \sin \left (x \right ) = y \ln \left (y\right )
\] |
[_separable] |
✓ |
8.257 |
|
|
\[
{}1+y^{2}+x y y^{\prime } = 0
\] |
[_separable] |
✓ |
2.773 |
|
|
\[
{}x y y^{\prime }-x y = y
\] |
[_quadrature] |
✓ |
0.546 |
|
|
\[
{}y^{\prime } = \frac {2 x y^{2}+x}{x^{2} y-y}
\] |
[_separable] |
✓ |
2.276 |
|
|
\[
{}y y^{\prime }+x y^{2}-8 x = 0
\] |
[_separable] |
✓ |
2.902 |
|
|
\[
{}y^{\prime }+2 x y^{2} = 0
\] |
[_separable] |
✓ |
1.976 |
|