|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+y x = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.435 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (-x^{2}+1\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.865 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y = 2
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.843 |
|
|
\[
{}y^{\prime \prime }+a \,x^{2} y = x +1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.554 |
|
|
\[
{}x^{4} y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.164 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.921 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.875 |
|
|
\[
{}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.937 |
|
|
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.933 |
|
|
\[
{}x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.858 |
|
|
\[
{}x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (-n +1\right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (-n +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.099 |
|
|
\[
{}x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.074 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+y^{\prime } x -n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.645 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.633 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.702 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+p x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.668 |
|
|
\[
{}x y^{\prime \prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.067 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.106 |
|
|
\[
{}x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.235 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.177 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-3 x^{2}+1\right ) y^{\prime }-y x = 0
\] |
[[_elliptic, _class_I]] |
✓ |
0.703 |
|
|
\[
{}y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
0.161 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.312 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+y x = 0
\] |
[[_elliptic, _class_II]] |
✓ |
0.710 |
|
|
\[
{}4 x \left (1-x \right ) y^{\prime \prime }-4 y^{\prime }-y = 0
\] |
[_Jacobi] |
✓ |
1.309 |
|
|
\[
{}x^{3} y^{\prime \prime }+y = x^{{3}/{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.120 |
|
|
\[
{}2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x} = \sqrt {x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.140 |
|
|
\[
{}\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y = 3 x^{2}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.000 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4} = 0
\] |
[_Jacobi] |
✓ |
0.853 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.221 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0
\] |
[_Jacobi] |
✓ |
0.838 |
|
|
\[
{}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9} = 0
\] |
[_Jacobi] |
✓ |
0.874 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+4 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.826 |
|
|
\[
{}4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.626 |
|
|
\[
{}y^{2}+y^{\prime } = \frac {a^{2}}{x^{4}}
\] |
[_rational, _Riccati] |
✓ |
1.436 |
|
|
\[
{}u^{\prime \prime }-\frac {a^{2} u}{x^{{2}/{3}}} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.772 |
|
|
\[
{}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.137 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.277 |
|
|
\[
{}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.265 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.480 |
|
|
\[
{}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.331 |
|
|
\[
{}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.204 |
|
|
\[
{}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.047 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -\left (x^{2}+\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.212 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.532 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {25}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.014 |
|
|
\[
{}y^{\prime \prime }+q y^{\prime } = \frac {2 y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.062 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{2 x} y = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.659 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{4 x} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.649 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.686 |
|
|
\[
{}x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.857 |
|
|
\[
{}y^{\prime } = y
\] |
[_quadrature] |
✓ |
0.433 |
|
|
\[
{}y^{\prime } x = y
\] |
[_separable] |
✓ |
1.324 |
|
|
\[
{}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
2.241 |
|
|
\[
{}\sin \left (x \right ) y^{\prime } = y \ln \left (y\right )
\] |
[_separable] |
✓ |
2.610 |
|
|
\[
{}1+y^{2}+x y y^{\prime } = 0
\] |
[_separable] |
✓ |
2.320 |
|
|
\[
{}x y y^{\prime }-y x = y
\] |
[_quadrature] |
✓ |
0.458 |
|
|
\[
{}y^{\prime } = \frac {2 x y^{2}+x}{x^{2} y-y}
\] |
[_separable] |
✓ |
1.760 |
|
|
\[
{}y y^{\prime }+x y^{2}-8 x = 0
\] |
[_separable] |
✓ |
1.574 |
|
|
\[
{}y^{\prime }+2 x y^{2} = 0
\] |
[_separable] |
✓ |
1.708 |
|
|
\[
{}\left (1+y\right ) y^{\prime } = y
\] |
[_quadrature] |
✓ |
0.628 |
|
|
\[
{}y^{\prime }-y x = x
\] |
[_separable] |
✓ |
1.230 |
|
|
\[
{}2 y^{\prime } = 3 \left (y-2\right )^{{1}/{3}}
\] |
[_quadrature] |
✓ |
0.571 |
|
|
\[
{}\left (x +y x \right ) y^{\prime }+y = 0
\] |
[_separable] |
✓ |
1.598 |
|
|
\[
{}y^{\prime }+y = {\mathrm e}^{x}
\] |
[[_linear, ‘class A‘]] |
✓ |
0.145 |
|
|
\[
{}x^{2} y^{\prime }+3 y x = 1
\] |
[_linear] |
✓ |
0.128 |
|
|
\[
{}y^{\prime }+2 y x -x \,{\mathrm e}^{-x^{2}} = 0
\] |
[_linear] |
✓ |
0.158 |
|
|
\[
{}2 y^{\prime } x +y = 2 x^{{5}/{2}}
\] |
[_linear] |
✓ |
0.138 |
|
|
\[
{}\cos \left (x \right ) y^{\prime }+y = \cos \left (x \right )^{2}
\] |
[_linear] |
✓ |
0.282 |
|
|
\[
{}y^{\prime }+\frac {y}{\sqrt {x^{2}+1}} = \frac {1}{x +\sqrt {x^{2}+1}}
\] |
[_linear] |
✓ |
0.169 |
|
|
\[
{}\left (1+{\mathrm e}^{x}\right ) y^{\prime }+2 y \,{\mathrm e}^{x} = \left (1+{\mathrm e}^{x}\right ) {\mathrm e}^{x}
\] |
[_linear] |
✓ |
0.175 |
|
|
\[
{}y^{\prime } x \ln \left (x \right )+y = \ln \left (x \right )
\] |
[_linear] |
✓ |
0.153 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime } = y x +2 x \sqrt {-x^{2}+1}
\] |
[_linear] |
✓ |
0.174 |
|
|
\[
{}y^{\prime }+y \tanh \left (x \right ) = 2 \,{\mathrm e}^{x}
\] |
[_linear] |
✓ |
0.194 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \sin \left (2 x \right )
\] |
[_linear] |
✓ |
0.212 |
|
|
\[
{}x^{\prime } = \cos \left (y \right )-x \tan \left (y \right )
\] |
[_linear] |
✓ |
0.178 |
|
|
\[
{}x^{\prime }+x-{\mathrm e}^{y} = 0
\] |
[[_linear, ‘class A‘]] |
✓ |
0.144 |
|
|
\[
{}x^{\prime } = \frac {3 y^{{2}/{3}}-x}{3 y}
\] |
[_linear] |
✓ |
0.123 |
|
|
\[
{}y^{\prime }+y = x y^{{2}/{3}}
\] |
[_Bernoulli] |
✓ |
1.228 |
|
|
\[
{}y^{\prime }+\frac {y}{x} = 2 x^{{3}/{2}} \sqrt {y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
9.130 |
|
|
\[
{}3 x y^{2} y^{\prime }+3 y^{3} = 1
\] |
[_separable] |
✓ |
2.382 |
|
|
\[
{}2 x \,{\mathrm e}^{3 y}+{\mathrm e}^{x}+\left (3 x^{2} {\mathrm e}^{3 y}-y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.642 |
|
|
\[
{}\left (x -y\right ) y^{\prime }+x +y+1 = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.372 |
|
|
\[
{}\cos \left (x \right ) \cos \left (y\right )+\sin \left (x \right )^{2}-\left (\sin \left (x \right ) \sin \left (y\right )+\cos \left (y\right )^{2}\right ) y^{\prime } = 0
\] |
unknown |
✓ |
77.923 |
|
|
\[
{}x^{2} y^{\prime }+y^{2}-y x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
1.846 |
|
|
\[
{}y y^{\prime } = -x +\sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.146 |
|
|
\[
{}y x +\left (y^{2}-x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.728 |
|
|
\[
{}y^{2}-y x +\left (y x +x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.808 |
|
|
\[
{}y^{\prime } = \cos \left (x +y\right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.061 |
|
|
\[
{}y^{\prime } = \frac {y}{x}-\tan \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.255 |
|
|
\[
{}\left (x -1\right ) y^{\prime }+y-\frac {1}{x^{2}}+\frac {2}{x^{3}} = 0
\] |
[_linear] |
✓ |
2.449 |
|
|
\[
{}y^{\prime } = x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.582 |
|
|
\[
{}y^{\prime } = \frac {2 y^{2}}{x}+\frac {y}{x}-2 x
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.063 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{-x} y^{2}+y-{\mathrm e}^{x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.168 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.734 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.758 |
|
|
\[
{}y^{\prime \prime }+9 y^{\prime } = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.257 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.161 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+6 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.621 |
|
|
\[
{}y^{\prime \prime }+16 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.731 |
|