|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.714 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
0.607 |
|
|
\[
{}\left (y^{2}+2 x^{2} y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} \left (x +y\right )+x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
152.296 |
|
|
\[
{}y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x \left (a^{2}-x^{2}\right )} = \frac {x^{2}}{a \left (a^{2}-x^{2}\right )}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.638 |
|
|
\[
{}\left (x^{3}+x +1\right ) y^{\prime \prime \prime }+\left (3+6 x \right ) y^{\prime \prime }+6 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✗ |
4.026 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+3 x^{2} y = 2 x
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.457 |
|
|
\[
{}y^{\prime \prime \prime \prime }-a^{2} y^{\prime \prime } = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.058 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime \prime } = n \sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
18.904 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y = \frac {2}{x^{3}}
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.408 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+{y^{\prime }}^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.225 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.299 |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.649 |
|
|
\[
{}x y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.335 |
|
|
\[
{}y^{\prime \prime } = \frac {a}{x}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.847 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.538 |
|
|
\[
{}y^{\prime \prime \prime } = \sin \left (x \right )^{2}
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.151 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } = {\mathrm e}^{x}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.849 |
|
|
\[
{}y^{\prime \prime \prime }+\cos \left (x \right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }-y \cos \left (x \right ) = \sin \left (2 x \right )
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.697 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2} = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.862 |
|
|
\[
{}a y^{\prime \prime } = y^{\prime }
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.202 |
|
|
\[
{}y^{3} y^{\prime \prime } = a
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.227 |
|
|
\[
{}y^{\prime \prime \prime } = f \left (x \right )
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.166 |
|
|
\[
{}y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.646 |
|
|
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-y = {\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.066 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.674 |
|
|
\[
{}\left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.168 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.239 |
|
|
\[
{}3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.228 |
|
|
\[
{}a^{2} {y^{\prime \prime }}^{2} = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
11.664 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{1}/{3}}}-\frac {6}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.765 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.408 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.881 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 \left (x^{2}+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.635 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.125 |
|
|
\[
{}y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+4 \csc \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.759 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.679 |
|
|
\[
{}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+a^{2} y = \frac {1}{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.364 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.195 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.954 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.472 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.458 |
|
|
\[
{}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+3 \left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.115 |
|
|
\[
{}y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.543 |
|
|
\[
{}y^{\prime \prime }+4 x y^{\prime }+4 x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.828 |
|
|
\[
{}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.099 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = x \left (-x^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.309 |
|
|
\[
{}\left (x \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.159 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.038 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.183 |
|
|
\[
{}y^{\prime \prime }+x y^{\prime }-y = f \left (x \right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.534 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.483 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2} y}{a} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.088 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.406 |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.131 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
2.309 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.133 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=0 \\ 5 x+y^{\prime }+3 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.532 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }-7 x+y=0 \\ y^{\prime }-2 x-5 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.544 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 x-3 y=t \\ y^{\prime }-3 x+2 y={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.519 |
|
|
\[
{}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+44 x+49 y=t \\ 3 x^{\prime }+7 y^{\prime }+34 x+38 y={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.643 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ y^{\prime \prime }+x+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.025 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 y^{\prime }-2 x+2 y=3 \,{\mathrm e}^{t} \\ 3 x^{\prime }+y^{\prime }+2 x+y=4 \,{\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.608 |
|
|
\[
{}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+2 x+31 y={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+x+24 y=3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.888 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+4 x+3 y=t \\ y^{\prime }+2 x+5 y={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.634 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=n y-m z \\ y^{\prime }=L z-m x \\ z^{\prime }=m x-L y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
108.300 |
|
|
\[
{}x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y = \frac {1}{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.665 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.473 |
|
|
\[
{}y+x +x y^{\prime } = 0
\] |
[_linear] |
✓ |
2.796 |
|
|
\[
{}y \left (1+x y\right )-x y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.466 |
|
|
\[
{}\sin \left (x \right ) y^{\prime }-y \cos \left (x \right )+y^{2} = 0
\] |
[_Bernoulli] |
✓ |
2.660 |
|
|
\[
{}\left (x +y\right ) y^{\prime }+y-x = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.395 |
|
|
\[
{}x +y y^{\prime }+\frac {-y+x y^{\prime }}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
1.912 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
27.130 |
|
|
\[
{}x y^{\prime }+y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.441 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }-2 x y = -x^{3}+x
\] |
[_linear] |
✓ |
1.874 |
|
|
\[
{}x y^{\prime }-y-\cos \left (\frac {1}{x}\right ) = 0
\] |
[_linear] |
✓ |
1.583 |
|
|
\[
{}x +y y^{\prime } = m \left (-y+x y^{\prime }\right )
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.369 |
|
|
\[
{}x \cos \left (y\right )^{2} = y \cos \left (x \right )^{2} y^{\prime }
\] |
[_separable] |
✓ |
4.627 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{x -y}+x^{2} {\mathrm e}^{-y}
\] |
[_separable] |
✓ |
1.569 |
|
|
\[
{}x^{2} y^{\prime }+y = 1
\] |
[_separable] |
✓ |
1.545 |
|
|
\[
{}2 y+\left (x^{2}+1\right ) \arctan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.066 |
|
|
\[
{}x y^{2}+x +\left (x^{2} y+y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.776 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{x +y}+x^{2} {\mathrm e}^{y}
\] |
[_separable] |
✓ |
1.552 |
|
|
\[
{}\left (3+2 \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime } = 1+2 \sin \left (y\right )+\cos \left (y\right )
\] |
[_separable] |
✓ |
9.441 |
|
|
\[
{}\frac {\cos \left (y\right )^{2} y^{\prime }}{x}+\frac {\cos \left (x \right )^{2}}{y} = 0
\] |
[_separable] |
✓ |
4.383 |
|
|
\[
{}\left ({\mathrm e}^{x}+1\right ) y y^{\prime } = \left (y+1\right ) {\mathrm e}^{x}
\] |
[_separable] |
✓ |
2.066 |
|
|
\[
{}\csc \left (x \right ) \ln \left (y\right ) y^{\prime }+y^{2} x^{2} = 0
\] |
[_separable] |
✓ |
6.791 |
|
|
\[
{}y^{\prime } = \frac {\sin \left (x \right )+x \cos \left (x \right )}{y \left (2 \ln \left (y\right )+1\right )}
\] |
[_separable] |
✓ |
36.338 |
|
|
\[
{}\cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right ) = \cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime }
\] |
[_separable] |
✓ |
27.982 |
|
|
\[
{}\left (x^{2}-x^{2} y\right ) y^{\prime }+y^{2}+x y^{2} = 0
\] |
[_separable] |
✓ |
1.700 |
|
|
\[
{}\left (\sin \left (y\right )+y \cos \left (y\right )\right ) y^{\prime }-\left (2 \ln \left (x \right )+1\right ) x = 0
\] |
[_separable] |
✓ |
35.467 |
|
|
\[
{}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
3.306 |
|
|
\[
{}y-x y^{\prime } = a \left (y^{2}+y^{\prime }\right )
\] |
[_separable] |
✓ |
1.605 |
|
|
\[
{}\left (x +y-1\right ) y^{\prime } = x +y+1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.825 |
|
|
\[
{}\left (2 x +2 y+1\right ) y^{\prime } = x +y+1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.000 |
|
|
\[
{}\left (2 x +3 y-5\right ) y^{\prime }+2 x +3 y-1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.923 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime } = x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.841 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y-\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.737 |
|
|
\[
{}x^{2}-y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
8.918 |
|
|
\[
{}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.996 |
|