|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}r^{\prime \prime } = -\frac {k}{r^{2}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
66.330 |
|
|
\[
{}y^{\prime \prime } = \frac {3 k y^{2}}{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
4.692 |
|
|
\[
{}y^{\prime \prime } = 2 k y^{3}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
5.739 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.860 |
|
|
\[
{}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.589 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{3}-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.487 |
|
|
\[
{}y y^{\prime \prime }-3 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.526 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
0.825 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (y^{\prime }+1\right ) = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.405 |
|
|
\[
{}\left (y+1\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.590 |
|
|
\[
{}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
2.502 |
|
|
\[
{}y^{\prime \prime } = 2 y y^{\prime }
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
13.908 |
|
|
\[
{}2 y^{\prime \prime } = {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.305 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime } = 1
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.089 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime } = x^{2}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.419 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+y y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.187 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.177 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.124 |
|
|
\[
{}-a y^{3}-\frac {b}{x^{{3}/{2}}}+y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
4.980 |
|
|
\[
{}a x y^{3}+b y^{2}+y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
2.475 |
|
|
\[
{}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0
\] |
[_Abel] |
✓ |
3.372 |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✗ |
108.821 |
|
|
\[
{}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0
\] |
[_rational, _Abel] |
✗ |
0.889 |
|
|
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✗ |
2.217 |
|
|
\[
{}y^{\prime }+y \tan \left (x \right ) = 0
\] |
[_separable] |
✓ |
1.794 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.979 |
|
|
\[
{}y {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.876 |
|
|
\[
{}\left (-x^{2}+1\right ) {y^{\prime }}^{2}+1 = 0
\] |
[_quadrature] |
✓ |
0.360 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{a x}+a y
\] |
[[_linear, ‘class A‘]] |
✓ |
0.767 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right )^{3} = a^{2} {y^{\prime \prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
205.148 |
|
|
\[
{}\left (x +1\right ) y+\left (1-y\right ) x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.495 |
|
|
\[
{}y^{\prime } = a y^{2} x
\] |
[_separable] |
✓ |
1.448 |
|
|
\[
{}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.760 |
|
|
\[
{}x y \left (x^{2}+1\right ) y^{\prime } = 1+y^{2}
\] |
[_separable] |
✓ |
4.317 |
|
|
\[
{}\frac {x}{y+1} = \frac {y y^{\prime }}{x +1}
\] |
[_separable] |
✓ |
1.596 |
|
|
\[
{}y^{\prime }+b^{2} y^{2} = a^{2}
\] |
[_quadrature] |
✓ |
3.583 |
|
|
\[
{}y^{\prime } = \frac {1+y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
2.274 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right ) = \cos \left (x \right ) \sin \left (y\right ) y^{\prime }
\] |
[_separable] |
✓ |
3.194 |
|
|
\[
{}a x y^{\prime }+2 y = x y y^{\prime }
\] |
[_separable] |
✓ |
1.662 |
|
|
\[
{}x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.774 |
|
|
\[
{}y^{\prime \prime }+x y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.228 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.698 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y = 2
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.644 |
|
|
\[
{}y^{\prime \prime }+a \,x^{2} y = x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.351 |
|
|
\[
{}x^{4} y^{\prime \prime }+x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.153 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.733 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.697 |
|
|
\[
{}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.783 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.704 |
|
|
\[
{}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.695 |
|
|
\[
{}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.900 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.828 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+x y^{\prime }-n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.533 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.518 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.497 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+p x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.444 |
|
|
\[
{}x y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.033 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.092 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.380 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.219 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-3 x^{2}+1\right ) y^{\prime }-x y = 0
\] |
[[_elliptic, _class_I]] |
✓ |
0.479 |
|
|
\[
{}y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
0.147 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.346 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+x y = 0
\] |
[[_elliptic, _class_II]] |
✓ |
0.487 |
|
|
\[
{}4 x \left (1-x \right ) y^{\prime \prime }-4 y^{\prime }-y = 0
\] |
[_Jacobi] |
✓ |
1.391 |
|
|
\[
{}x^{3} y^{\prime \prime }+y = x^{{3}/{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.105 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = \sqrt {x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.125 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 3 x^{2}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.798 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0
\] |
[_Jacobi] |
✓ |
0.725 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.336 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0
\] |
[_Jacobi] |
✓ |
0.726 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9} = 0
\] |
[_Jacobi] |
✓ |
0.737 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+4 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.697 |
|
|
\[
{}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.432 |
|
|
\[
{}y^{\prime }+y^{2} = \frac {a^{2}}{x^{4}}
\] |
[_rational, _Riccati] |
✓ |
1.638 |
|
|
\[
{}u^{\prime \prime }-\frac {a^{2} u}{x^{{2}/{3}}} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.667 |
|
|
\[
{}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.165 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.175 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.287 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.815 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.663 |
|
|
\[
{}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.316 |
|
|
\[
{}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.208 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.106 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.144 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.899 |
|
|
\[
{}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.959 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{2 x} y = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.453 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{4 x} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.574 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.493 |
|
|
\[
{}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.104 |
|
|
\[
{}y^{\prime } = y
\] |
[_quadrature] |
✓ |
1.528 |
|
|
\[
{}x y^{\prime } = y
\] |
[_separable] |
✓ |
2.029 |
|
|
\[
{}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
7.838 |
|
|
\[
{}\sin \left (x \right ) y^{\prime } = y \ln \left (y\right )
\] |
[_separable] |
✓ |
16.326 |
|
|
\[
{}1+y^{2}+x y y^{\prime } = 0
\] |
[_separable] |
✓ |
3.221 |
|
|
\[
{}x y y^{\prime }-x y = y
\] |
[_quadrature] |
✓ |
0.962 |
|
|
\[
{}y^{\prime } = \frac {2 x y^{2}+x}{x^{2} y-y}
\] |
[_separable] |
✓ |
2.790 |
|
|
\[
{}y y^{\prime }+x y^{2}-8 x = 0
\] |
[_separable] |
✓ |
3.589 |
|
|
\[
{}y^{\prime }+2 x y^{2} = 0
\] |
[_separable] |
✓ |
2.492 |
|