|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.402 |
|
|
\[
{}y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.140 |
|
|
\[
{}y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.590 |
|
|
\[
{}y^{\prime \prime } y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.718 |
|
|
\[
{}y^{\prime \prime } y^{\prime }+y^{n} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
3.084 |
|
|
\[
{}y^{\prime } = \left (x +y\right )^{4}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
148.632 |
|
|
\[
{}y^{\prime \prime }+\left (x +3\right ) y^{\prime }+\left (y^{2}+3\right ) {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.140 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }+y {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.305 |
|
|
\[
{}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.115 |
|
|
\[
{}3 y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.582 |
|
|
\[
{}10 y^{\prime \prime }+x^{2} y^{\prime }+\frac {3 {y^{\prime }}^{2}}{y} = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.307 |
|
|
\[
{}10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.921 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y}{x^{2}} = x \,{\mathrm e}^{-\sqrt {x}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.130 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.695 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.324 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-c^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.046 |
|
|
\[
{}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+a^{2} y = \frac {1}{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.172 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y = 2 x^{3}-x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.960 |
|
|
\[
{}y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+4 y \csc \left (x \right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
8.423 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y = 4 \cos \left (\ln \left (x +1\right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
1.417 |
|
|
\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.922 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = 8 x^{3} \sin \left (x \right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
27.496 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.608 |
|
|
\[
{}\cos \left (x \right ) y^{\prime \prime }+\sin \left (x \right ) y^{\prime }-2 y \cos \left (x \right )^{3} = 2 \cos \left (x \right )^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.705 |
|
|
\[
{}y^{\prime \prime }+\left (1-\frac {1}{x}\right ) y^{\prime }+4 x^{2} y \,{\mathrm e}^{-2 x} = 4 \left (x^{3}+x^{2}\right ) {\mathrm e}^{-3 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.757 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = x^{m +1}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.931 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.281 |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\cos \left (x \right )^{2} y = 0
\] |
[_Lienard] |
✓ |
3.157 |
|
|
\[
{}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y = -3 \,{\mathrm e}^{x^{2}} \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.694 |
|
|
\[
{}y^{\prime \prime }-2 b x y^{\prime }+b^{2} x^{2} y = x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.674 |
|
|
\[
{}y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-3\right ) y = {\mathrm e}^{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.262 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = {\mathrm e}^{x^{2}} \sec \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
15.432 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 \left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.253 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.553 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.148 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }-x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.177 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+x y = 0
\] |
[_Lienard] |
✓ |
1.521 |
|
|
\[
{}y^{\prime }+y \cot \left (x \right ) = 2 \cos \left (x \right )
\] |
[_linear] |
✓ |
1.677 |
|
|
\[
{}2 x y^{2}-y+\left (y^{2}+x +y\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.369 |
|
|
\[
{}y^{\prime } = x -y^{2}
\] |
[[_Riccati, _special]] |
✓ |
0.925 |
|
|
\[
{}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x}
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.194 |
|
|
\[
{}x^{2} y^{\prime \prime }-x \left (x +6\right ) y^{\prime }+10 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.383 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-5\right ) y = 0
\] |
[_Bessel] |
✓ |
0.844 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-5\right ) y = 0
\] |
[_Bessel] |
✓ |
2.165 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.173 |
|
|
\[
{}y^{\prime \prime \prime }-x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.041 |
|
|
\[
{}y^{\prime } = y^{{1}/{3}}
\] |
[_quadrature] |
✓ |
1.676 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=3 x+y \\ y^{\prime }=y-x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.284 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[_Gegenbauer] |
✓ |
0.256 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y = 0
\] |
[_Gegenbauer] |
✓ |
0.266 |
|
|
\[
{}\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.480 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+8 x y^{\prime }+12 y = 0
\] |
[_Gegenbauer] |
✓ |
0.237 |
|
|
\[
{}3 y^{\prime \prime }+x y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.282 |
|
|
\[
{}5 y^{\prime \prime }-2 x y^{\prime }+10 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.319 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-3 x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.310 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.341 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.273 |
|
|
\[
{}\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.349 |
|
|
\[
{}\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.276 |
|
|
\[
{}t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.381 |
|
|
\[
{}t^{2} y^{\prime \prime }-t \left (t +2\right ) y^{\prime }+\left (t +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.133 |
|
|
\[
{}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0
\] |
[_Laguerre] |
✓ |
0.275 |
|
|
\[
{}\left (-t +1\right ) y^{\prime \prime }+t y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.295 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.201 |
|
|
\[
{}t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y = 0
\] |
[_Laguerre] |
✓ |
0.313 |
|
|
\[
{}\left (-t +1\right ) y^{\prime \prime }+t y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.329 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.280 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.313 |
|
|
\[
{}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.280 |
|
|
\[
{}2 y^{\prime \prime }+x y^{\prime }+3 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.287 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.279 |
|
|
\[
{}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.291 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.280 |
|
|
\[
{}\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.066 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.145 |
|
|
\[
{}\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.299 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.187 |
|
|
\[
{}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.381 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.153 |
|
|
\[
{}y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.116 |
|
|
\[
{}\left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.281 |
|
|
\[
{}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.186 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.188 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.145 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.203 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.121 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.127 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +2\right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.132 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.195 |
|
|
\[
{}\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.290 |
|
|
\[
{}4 x^{2} y^{\prime \prime }-4 x \left (x +1\right ) y^{\prime }+\left (2 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.155 |
|
|
\[
{}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }-\left (6 x -8\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.304 |
|
|
\[
{}\left (x +2\right ) y^{\prime \prime }+x y^{\prime }+3 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.359 |
|
|
\[
{}x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (4+x \right ) y^{\prime }+\left (2-x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.298 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-\left (4+6 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.301 |
|
|
\[
{}x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.421 |
|
|
\[
{}x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.535 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.332 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.346 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.336 |
|