|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime \prime } = f \left (x \right )
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.389 |
|
|
\[
{}y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
14.529 |
|
|
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-y = {\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.184 |
|
|
\[
{}y^{\prime \prime }-x^{2} y^{\prime }+x y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.667 |
|
|
\[
{}\left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.157 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.240 |
|
|
\[
{}3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.362 |
|
|
\[
{}a^{2} {y^{\prime \prime }}^{2} = {y^{\prime }}^{2}+1
\] |
[[_2nd_order, _missing_x]] |
✓ |
11.625 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{1}/{3}}}-\frac {6}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.613 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.509 |
|
|
\[
{}y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }+5 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.992 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 \left (x^{2}+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.821 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.219 |
|
|
\[
{}y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+4 \csc \left (x \right )^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.758 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.245 |
|
|
\[
{}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+a^{2} y = \frac {1}{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.944 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x} = n^{2} y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.299 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.176 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.156 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+3 y^{\prime } x +y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.433 |
|
|
\[
{}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+3 \left (x -2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.337 |
|
|
\[
{}y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.526 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +4 x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.004 |
|
|
\[
{}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.319 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x -y = x \left (-x^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.062 |
|
|
\[
{}\left (x \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+y \cos \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.380 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.059 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x -a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.412 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x -y = f \left (x \right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.544 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.705 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2} y}{a} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.813 |
|
|
\[
{}\left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.356 |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.151 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.225 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.223 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=0 \\ 5 x+y^{\prime }+3 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.477 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }-7 x+y=0 \\ y^{\prime }-2 x-5 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.495 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 x-3 y=t \\ y^{\prime }-3 x+2 y={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.491 |
|
|
\[
{}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+44 x+49 y=t \\ 3 x^{\prime }+7 y^{\prime }+34 x+38 y={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.534 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ y^{\prime \prime }+x+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.052 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+2 y^{\prime }-2 x+2 y=3 \,{\mathrm e}^{t} \\ 3 x^{\prime }+y^{\prime }+2 x+y=4 \,{\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.515 |
|
|
\[
{}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+2 x+31 y={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+x+24 y=3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.786 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }+4 x+3 y=t \\ y^{\prime }+2 x+5 y={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.536 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=n y-m z \\ y^{\prime }=L z-m x \\ z^{\prime }=m x-L y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
93.145 |
|
|
\[
{}x^{2} y^{\prime \prime }-5 y^{\prime } x +5 y = \frac {1}{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.750 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{r} = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.663 |
|
|
\[
{}y+x +y^{\prime } x = 0
\] |
[_linear] |
✓ |
2.371 |
|
|
\[
{}y \left (1+x y\right )-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.131 |
|
|
\[
{}\sin \left (x \right ) y^{\prime }-y \cos \left (x \right )+y^{2} = 0
\] |
[_Bernoulli] |
✓ |
2.369 |
|
|
\[
{}\left (x +y\right ) y^{\prime }+y-x = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.580 |
|
|
\[
{}x +y y^{\prime }+\frac {-y+y^{\prime } x}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
1.745 |
|
|
\[
{}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
70.332 |
|
|
\[
{}y^{\prime } x +y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.027 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }-2 x y = -x^{3}+x
\] |
[_linear] |
✓ |
1.723 |
|
|
\[
{}y^{\prime } x -y-\cos \left (\frac {1}{x}\right ) = 0
\] |
[_linear] |
✓ |
1.505 |
|
|
\[
{}x +y y^{\prime } = m \left (-y+y^{\prime } x \right )
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.779 |
|
|
\[
{}x \cos \left (y\right )^{2} = y \cos \left (x \right )^{2} y^{\prime }
\] |
[_separable] |
✓ |
3.944 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{x -y}+x^{2} {\mathrm e}^{-y}
\] |
[_separable] |
✓ |
1.408 |
|
|
\[
{}x^{2} y^{\prime }+y = 1
\] |
[_separable] |
✓ |
1.519 |
|
|
\[
{}2 y+\left (x^{2}+1\right ) \arctan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.003 |
|
|
\[
{}x y^{2}+x +\left (x^{2} y+y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.980 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{x +y}+x^{2} {\mathrm e}^{y}
\] |
[_separable] |
✓ |
1.311 |
|
|
\[
{}\left (3+2 \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime } = 1+2 \sin \left (y\right )+\cos \left (y\right )
\] |
[_separable] |
✓ |
9.122 |
|
|
\[
{}\frac {\cos \left (y\right )^{2} y^{\prime }}{x}+\frac {\cos \left (x \right )^{2}}{y} = 0
\] |
[_separable] |
✓ |
3.706 |
|
|
\[
{}\left ({\mathrm e}^{x}+1\right ) y y^{\prime } = \left (1+y\right ) {\mathrm e}^{x}
\] |
[_separable] |
✓ |
1.797 |
|
|
\[
{}\csc \left (x \right ) \ln \left (y\right ) y^{\prime }+x^{2} y^{2} = 0
\] |
[_separable] |
✓ |
5.914 |
|
|
\[
{}y^{\prime } = \frac {\sin \left (x \right )+x \cos \left (x \right )}{y \left (2 \ln \left (y\right )+1\right )}
\] |
[_separable] |
✓ |
35.648 |
|
|
\[
{}\cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right ) = \cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime }
\] |
[_separable] |
✓ |
22.421 |
|
|
\[
{}\left (x^{2}-x^{2} y\right ) y^{\prime }+y^{2}+x y^{2} = 0
\] |
[_separable] |
✓ |
1.660 |
|
|
\[
{}\left (\sin \left (y\right )+y \cos \left (y\right )\right ) y^{\prime }-\left (2 \ln \left (x \right )+1\right ) x = 0
\] |
[_separable] |
✓ |
34.685 |
|
|
\[
{}3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
2.980 |
|
|
\[
{}y-y^{\prime } x = a \left (y^{2}+y^{\prime }\right )
\] |
[_separable] |
✓ |
1.211 |
|
|
\[
{}\left (x +y-1\right ) y^{\prime } = x +y+1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.726 |
|
|
\[
{}\left (2 x +2 y+1\right ) y^{\prime } = x +y+1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.873 |
|
|
\[
{}\left (2 x +3 y-5\right ) y^{\prime }+2 x +3 y-1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.791 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime } = x y+x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.885 |
|
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y-\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
6.836 |
|
|
\[
{}x^{2}-y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
6.973 |
|
|
\[
{}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.403 |
|
|
\[
{}\left (2 x -2 y+5\right ) y^{\prime }-x +y-3 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.782 |
|
|
\[
{}x +y+1-\left (2 x +2 y+1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.877 |
|
|
\[
{}y^{2} = \left (x y-x^{2}\right ) y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
37.162 |
|
|
\[
{}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )-x
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.084 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime } = x y
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
3.388 |
|
|
\[
{}x^{2} y^{\prime }+y \left (x +y\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
3.175 |
|
|
\[
{}2 y^{\prime } = \frac {y}{x}+\frac {y^{2}}{x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
3.154 |
|
|
\[
{}\left (6 x -5 y+4\right ) y^{\prime }+y-2 x -1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
53.572 |
|
|
\[
{}\left (x -3 y+4\right ) y^{\prime }+7 y-5 x = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.147 |
|
|
\[
{}\left (2 x +4 y+3\right ) y^{\prime } = 2 y+x +1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.806 |
|
|
\[
{}-y+y^{\prime } x = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
6.288 |
|
|
\[
{}x \left (x^{2}+3 y^{2}\right )+y \left (y^{2}+3 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
68.118 |
|
|
\[
{}x^{2}+3 y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
4.276 |
|
|
\[
{}y^{\prime } = \frac {2 x -y+1}{x +2 y-3}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.457 |
|
|
\[
{}\left (x -y\right ) y^{\prime } = x +y+1
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.385 |
|
|
\[
{}x -y-2-\left (2 x -2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.798 |
|
|
\[
{}y^{\prime }+y \cot \left (x \right ) = 2 \cos \left (x \right )
\] |
[_linear] |
✓ |
1.769 |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime }+y = \tan \left (x \right )
\] |
[_linear] |
✓ |
3.744 |
|
|
\[
{}x \cos \left (x \right ) y^{\prime }+y \left (x \sin \left (x \right )+\cos \left (x \right )\right ) = 1
\] |
[_linear] |
✓ |
5.629 |
|
|
\[
{}y-x \sin \left (x^{2}\right )+y^{\prime } x = 0
\] |
[_linear] |
✓ |
1.482 |
|
|
\[
{}x \ln \left (x \right ) y^{\prime }+y = 2 \ln \left (x \right )
\] |
[_linear] |
✓ |
1.057 |
|