|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}a^{2} y^{\prime \prime } y^{\prime } = x
\] |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
1.444 |
|
|
\[
{}y^{\prime \prime \prime } y^{\prime \prime } = 2
\] |
[[_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]] |
✓ |
0.446 |
|
|
\[
{}y^{\prime \prime } = {y^{\prime }}^{2}+1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
4.561 |
|
|
\[
{}a y^{\prime \prime } = \sqrt {{y^{\prime }}^{2}+1}
\] |
[[_2nd_order, _missing_x]] |
✓ |
4.095 |
|
|
\[
{}y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
14.753 |
|
|
\[
{}a^{2} {y^{\prime \prime }}^{2} = {y^{\prime }}^{2}+1
\] |
[[_2nd_order, _missing_x]] |
✓ |
11.658 |
|
|
\[
{}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
3.901 |
|
|
\[
{}y^{\prime } = x y^{\prime \prime }+\sqrt {{y^{\prime }}^{2}+1}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
145.103 |
|
|
\[
{}a^{2} y^{\prime \prime \prime \prime } = y^{\prime \prime }
\] |
[[_high_order, _missing_x]] |
✓ |
0.096 |
|
|
\[
{}y^{\prime \prime \prime \prime }+a^{2} y^{\prime \prime } = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.079 |
|
|
\[
{}y^{\left (5\right )}-n^{2} y^{\prime \prime \prime } = {\mathrm e}^{a x}
\] |
[[_high_order, _missing_y]] |
✓ |
0.130 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+a^{2} y^{\prime \prime } = 0
\] |
[[_high_order, _missing_y]] |
✓ |
0.492 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime } = \lambda y^{\prime \prime }
\] |
[[_high_order, _missing_y]] |
✓ |
0.451 |
|
|
\[
{}n \,x^{3} y^{\prime \prime \prime } = y-y^{\prime } x
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.131 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2} = 3 y y^{\prime }
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.789 |
|
|
\[
{}2 x^{2} y y^{\prime \prime }+y^{2} = x^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.142 |
|
|
\[
{}x^{2} y^{\prime \prime } = \sqrt {m \,x^{2} {y^{\prime }}^{3}+n y^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.272 |
|
|
\[
{}x^{4} y^{\prime \prime } = \left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.148 |
|
|
\[
{}x^{4} y^{\prime \prime }-x^{3} y^{\prime } = x^{2} {y^{\prime }}^{2}-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.156 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 y^{2}-6 y = x^{4} {y^{\prime }}^{2}
\] |
[NONE] |
✗ |
0.146 |
|
|
\[
{}y^{\prime \prime } = {\mathrm e}^{y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
34.138 |
|
|
\[
{}y^{\prime \prime }+a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.734 |
|
|
\[
{}a y^{\prime \prime \prime } = y^{\prime \prime }
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.069 |
|
|
\[
{}x^{2} y^{\prime \prime \prime \prime }+1 = 0
\] |
[[_high_order, _quadrature]] |
✓ |
0.218 |
|
|
\[
{}y^{\prime \prime \prime } = \sin \left (x \right )^{2}
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.153 |
|
|
\[
{}y^{\prime \prime } = \frac {1}{\sqrt {a y}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.037 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+3 y^{\prime } x +y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.458 |
|
|
\[
{}-a y^{\prime \prime } = \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
8.239 |
|
|
\[
{}\sin \left (y\right )^{3} y^{\prime \prime } = \cos \left (y\right )
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
148.056 |
|
|
\[
{}{\mathrm e}^{x} \left (x y^{\prime \prime }-y^{\prime }\right ) = x^{3}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.115 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x = 2
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.391 |
|
|
\[
{}2 x y^{\prime \prime \prime } y^{\prime \prime } = {y^{\prime \prime }}^{2}-a^{2}
\] |
[[_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]] |
✓ |
32.324 |
|
|
\[
{}\left (x^{3}-4 x \right ) y^{\prime \prime \prime }+\left (9 x^{2}-4\right ) y^{\prime \prime }+18 y^{\prime } x +6 y = 6
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.221 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.741 |
|
|
\[
{}x y^{\prime \prime }-\left (x +3\right ) y^{\prime }+3 y = 0
\] |
[_Laguerre] |
✓ |
0.981 |
|
|
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime } = y+{\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.546 |
|
|
\[
{}\left (x +1\right ) y^{\prime \prime }-2 \left (x +3\right ) y^{\prime }+\left (x +5\right ) y = {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.189 |
|
|
\[
{}\left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.203 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x -y = X
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.070 |
|
|
\[
{}y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime }+x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.056 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
0.125 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+2 x \right ) y^{\prime }+\left (x +2\right ) y = x^{3} {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.415 |
|
|
\[
{}y^{\prime \prime }-a x y^{\prime }+a^{2} \left (x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.357 |
|
|
\[
{}\left (2 x^{3}-a \right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.367 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +4 x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.023 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.066 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.369 |
|
|
\[
{}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
32.475 |
|
|
\[
{}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
50.816 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{3}+6 x^{2}+4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.865 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.207 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = \sec \left (x \right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
15.447 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }-\left (a^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.665 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.148 |
|
|
\[
{}y^{\prime \prime }+2 n \cot \left (n x \right ) y^{\prime }+\left (m^{2}-n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.874 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {y \left (-8+\sqrt {x}+x \right )}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.263 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 n x y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.118 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.231 |
|
|
\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.542 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.305 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.457 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +m^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.583 |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\sin \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.630 |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }+\sin \left (x \right ) \cos \left (x \right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.038 |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.327 |
|
|
\[
{}y^{\prime \prime }+\left (\tan \left (x \right )-1\right )^{2} y^{\prime }-n \left (n -1\right ) y \sec \left (x \right )^{4} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.004 |
|
|
\[
{}y^{\prime \prime }+\left (3 \sin \left (x \right )-\cot \left (x \right )\right ) y^{\prime }+2 \sin \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.981 |
|
|
\[
{}3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.463 |
|
|
\[
{}x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-2 y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.254 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime }-\left (x^{2}+1\right ) y = {\mathrm e}^{-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.321 |
|
|
\[
{}\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.522 |
|
|
\[
{}y^{\prime \prime }+y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.962 |
|
|
\[
{}y^{\prime \prime }+y = \csc \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.886 |
|
|
\[
{}y^{\prime \prime }+4 y = 4 \tan \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
7.687 |
|
|
\[
{}\left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y = \left (1-x \right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.586 |
|
|
\[
{}y^{\prime \prime }-y = \frac {2}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.609 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-4 y^{\prime } x -\left (x^{2}+1\right ) y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.683 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = -4 x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.778 |
|
|
\[
{}-y+y^{\prime } x = \left (x -1\right ) \left (y^{\prime \prime }-x +1\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.586 |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.144 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.803 |
|
|
\[
{}\left (x^{2}+a \right ) y^{\prime \prime }-2 y^{\prime } x +2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.311 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.724 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime } x +\left (x^{2}+1\right ) y = x^{3}+3 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.858 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2} y}{a} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.981 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.318 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +\frac {a^{2} y}{-x^{2}+1} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.605 |
|
|
\[
{}\left (2 x -1\right ) y^{\prime \prime }-2 y^{\prime }+\left (3-2 x \right ) y = 2 \,{\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.608 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -y = 8 x^{3}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.828 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime } x +\left (x^{2}+5\right ) y = x \,{\mathrm e}^{-\frac {x^{2}}{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.772 |
|
|
\[
{}x \left (-x^{2}+1\right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) \left (3 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.111 |
|
|
\[
{}y^{\prime \prime }+\left (1-\frac {2}{x^{2}}\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.820 |
|
|
\[
{}\left (x^{3}-2 x^{2}\right ) y^{\prime \prime }+2 x^{2} y^{\prime }-12 \left (x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.953 |
|
|
\[
{}x y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+\left (x +2\right ) y = \left (x -2\right ) {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.994 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.282 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -9 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.310 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x -a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.065 |
|
|
\[
{}x y^{\prime \prime } \left (x \cos \left (x \right )-2 \sin \left (x \right )\right )+\left (x^{2}+2\right ) y^{\prime } \sin \left (x \right )-2 y \left (x \sin \left (x \right )+\cos \left (x \right )\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.421 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.601 |
|