|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=5 x_{1}-3 x_{2}-2 x_{3} \\ x_{2}^{\prime }=8 x_{1}-5 x_{2}-4 x_{3} \\ x_{3}^{\prime }=-4 x_{1}+3 x_{2}+3 x_{3} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.454 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-7 x_{1}+9 x_{2}-6 x_{3} \\ x_{2}^{\prime }=-8 x_{1}+11 x_{2}-7 x_{3} \\ x_{3}^{\prime }=-2 x_{1}+3 x_{2}-x_{3} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.516 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=5 x_{1}+6 x_{2}+2 x_{3} \\ x_{2}^{\prime }=-2 x_{1}-2 x_{2}-x_{3} \\ x_{3}^{\prime }=-2 x_{1}-3 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.424 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-8 x_{1}-16 x_{2}-16 x_{3}-17 x_{4} \\ x_{2}^{\prime }=-2 x_{1}-10 x_{2}-8 x_{3}-7 x_{4} \\ x_{3}^{\prime }=-2 x_{1}-2 x_{3}-3 x_{4} \\ x_{4}^{\prime }=6 x_{1}+14 x_{2}+14 x_{3}+14 x_{4} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
1.350 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}-x_{2}-2 x_{3}+3 x_{4} \\ x_{2}^{\prime }=2 x_{1}-\frac {3 x_{2}}{2}-x_{3}+\frac {7 x_{4}}{2} \\ x_{3}^{\prime }=-x_{1}+\frac {x_{2}}{2}-\frac {3 x_{4}}{2} \\ x_{4}^{\prime }=-2 x_{1}+\frac {3 x_{2}}{2}+3 x_{3}-\frac {7 x_{4}}{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.646 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}-4 x_{2} \\ x_{2}^{\prime }=4 x_{1}-7 x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.569 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=3 x_{1}-4 x_{2} \\ x_{2}^{\prime }=x_{1}-x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.561 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=4 x_{1}+x_{2}+3 x_{3} \\ x_{2}^{\prime }=6 x_{1}+4 x_{2}+6 x_{3} \\ x_{3}^{\prime }=-5 x_{1}-2 x_{2}-4 x_{3} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.486 |
|
|
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=x_{1}+x_{2} \\ x_{2}^{\prime }=-14 x_{1}-5 x_{2}+x_{3} \\ x_{3}^{\prime }=15 x_{1}+5 x_{2}-2 x_{3} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
0.487 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=-2 y+x y \\ y^{\prime }=x+4 x y \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.031 |
|
|
\[
{}\left [\begin {array}{c} x^{\prime }=1+5 y \\ y^{\prime }=1-6 x^{2} \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.031 |
|
|
\[
{}y^{\prime } = 2
\] |
[_quadrature] |
✓ |
0.801 |
|
|
\[
{}y^{\prime } = -x^{3}
\] |
[_quadrature] |
✓ |
0.453 |
|
|
\[
{}y^{\prime \prime } = \sin \left (x \right )
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.892 |
|
|
\[
{}x \sqrt {1+y^{2}}+y \sqrt {x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
38.145 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
37.098 |
|
|
\[
{}\sqrt {-x^{2}+1}\, y^{\prime }+\sqrt {1-y^{2}} = 0
\] |
[_separable] |
✓ |
40.595 |
|
|
\[
{}y^{\prime } = \frac {2 x y}{x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.888 |
|
|
\[
{}y^{\prime } = \frac {y \left (1+\ln \left (y\right )-\ln \left (x \right )\right )}{x}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.868 |
|
|
\[
{}y^{2}+x^{2} y^{\prime } = x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
40.454 |
|
|
\[
{}\left (x +y\right ) y^{\prime } = y-x
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
38.960 |
|
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.399 |
|
|
\[
{}3 y-7 x +7 = \left (3 x -7 y-3\right ) y^{\prime }
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.648 |
|
|
\[
{}\left (x +2 y+1\right ) y^{\prime } = 2 x +4 y+3
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.937 |
|
|
\[
{}y^{\prime } = \frac {2 \left (y+2\right )^{2}}{\left (x +y-1\right )^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
1.840 |
|
|
\[
{}\left (x +y\right )^{2} y^{\prime } = a^{2}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
4.213 |
|
|
\[
{}x y^{\prime }-4 y = x^{2} \sqrt {y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.931 |
|
|
\[
{}\cos \left (x \right ) y^{\prime } = y \sin \left (x \right )+\cos \left (x \right )^{2}
\] |
[_linear] |
✓ |
2.537 |
|
|
\[
{}y^{\prime } = 2 x y-x^{3}+x
\] |
[_linear] |
✓ |
1.667 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.389 |
|
|
\[
{}\left (x -2 x y-y^{2}\right ) y^{\prime }+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.106 |
|
|
\[
{}x y^{\prime }+y = x y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.371 |
|
|
\[
{}y^{\prime }-\frac {x y}{2 x^{2}-2}-\frac {x}{2 y} = 0
\] |
[_rational, _Bernoulli] |
✓ |
2.224 |
|
|
\[
{}y^{\prime } \left (x^{2} y^{3}+x y\right ) = 1
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.643 |
|
|
\[
{}x -y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.419 |
|
|
\[
{}y^{\prime } = \frac {y^{2}}{3}+\frac {2}{3 x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
1.887 |
|
|
\[
{}y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.258 |
|
|
\[
{}x y^{\prime }-3 y+y^{2} = 4 x^{2}-4 x
\] |
[_rational, _Riccati] |
✓ |
1.779 |
|
|
\[
{}y^{\prime } = y^{2}+\frac {1}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
1.589 |
|
|
\[
{}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime } = \left (1+y^{2}\right )^{{3}/{2}}
\] |
[‘y=_G(x,y’)‘] |
✗ |
221.525 |
|
|
\[
{}y^{\prime } \left (x^{2}+y^{2}+3\right ) = 2 x \left (2 y-\frac {x^{2}}{y}\right )
\] |
[_rational] |
✗ |
2.917 |
|
|
\[
{}y^{\prime } = \frac {x -y^{2}}{2 y \left (x +y^{2}\right )}
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.411 |
|
|
\[
{}\left (\left (x +y\right ) x +a^{2}\right ) y^{\prime } = y \left (x +y\right )+b^{2}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.587 |
|
|
\[
{}y^{\prime } = k y+f \left (x \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.049 |
|
|
\[
{}y^{\prime } = y^{2}-x^{2}
\] |
[_Riccati] |
✓ |
1.136 |
|
|
\[
{}\frac {x +y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y-x y^{\prime }}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
2.936 |
|
|
\[
{}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
34.625 |
|
|
\[
{}\frac {\sin \left (\frac {x}{y}\right )}{y}-\frac {y \cos \left (\frac {y}{x}\right )}{x^{2}}+1+\left (\frac {\cos \left (\frac {y}{x}\right )}{x}-\frac {x \sin \left (\frac {x}{y}\right )}{y^{2}}+\frac {1}{y^{2}}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
39.342 |
|
|
\[
{}\frac {1}{x}-\frac {y^{2}}{\left (x -y\right )^{2}}+\left (\frac {x^{2}}{\left (x -y\right )^{2}}-\frac {1}{y}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.946 |
|
|
\[
{}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.016 |
|
|
\[
{}\left (y^{2} x^{2}-1\right ) y^{\prime }+2 x y^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
7.622 |
|
|
\[
{}a x y^{\prime }+b y+x^{m} y^{n} \left (\alpha x y^{\prime }+\beta y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.892 |
|
|
\[
{}2 x y^{2}-y+\left (y^{2}+x +y\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.502 |
|
|
\[
{}y^{\prime } = 2 x y-x^{3}+x
\] |
[_linear] |
✓ |
1.664 |
|
|
\[
{}x y^{\prime }+y-x y^{2} \ln \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.367 |
|
|
\[
{}2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
2.671 |
|
|
\[
{}{y^{\prime }}^{2} y+y^{\prime } \left (x -y\right )-x = 0
\] |
[_quadrature] |
✓ |
0.474 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = y^{2} x^{2}+x^{4}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
9.706 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x^{3} y+y^{2} x^{2}+x y^{3}\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
1.109 |
|
|
\[
{}x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
12.278 |
|
|
\[
{}x {y^{\prime }}^{3} = 1+y^{\prime }
\] |
[_quadrature] |
✓ |
0.543 |
|
|
\[
{}{y^{\prime }}^{3}-x^{3} \left (1-y^{\prime }\right ) = 0
\] |
[_quadrature] |
✓ |
0.587 |
|
|
\[
{}{y^{\prime }}^{3}+y^{3}-3 y y^{\prime } = 0
\] |
[_quadrature] |
✓ |
177.885 |
|
|
\[
{}y = {\mathrm e}^{y^{\prime }} {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.724 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
1.591 |
|
|
\[
{}y \left (1+{y^{\prime }}^{2}\right ) = 2 \alpha
\] |
[_quadrature] |
✓ |
0.599 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (x y^{\prime }-2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.722 |
|
|
\[
{}y = 2 x y^{\prime }+\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
1.849 |
|
|
\[
{}y = \frac {k \left (x +y y^{\prime }\right )}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
9.878 |
|
|
\[
{}x = y y^{\prime }+a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✓ |
73.238 |
|
|
\[
{}y = x {y^{\prime }}^{2}+{y^{\prime }}^{3}
\] |
[_dAlembert] |
✓ |
3.024 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.420 |
|
|
\[
{}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
107.730 |
|
|
\[
{}{y^{\prime }}^{2} \left (x^{2}-1\right )-2 x y y^{\prime }+y^{2}-1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.597 |
|
|
\[
{}{y^{\prime }}^{2}+2 x y^{\prime }+2 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.482 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.165 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}+1
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.326 |
|
|
\[
{}y^{\prime } = \sqrt {y}
\] |
[_quadrature] |
✓ |
1.806 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )
\] |
[_quadrature] |
✓ |
1.555 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )^{2}
\] |
[_quadrature] |
✓ |
1.347 |
|
|
\[
{}y^{\prime } = -x +\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.502 |
|
|
\[
{}y^{\prime } = -x -\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.414 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.451 |
|
|
\[
{}x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
13.428 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
1.667 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (x y^{\prime }-2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.740 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.733 |
|
|
\[
{}y = {y^{\prime }}^{2}-x y^{\prime }+\frac {x^{3}}{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
5.742 |
|
|
\[
{}y = 2 x y^{\prime }+\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
1.975 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
4.035 |
|
|
\[
{}{y^{\prime \prime \prime }}^{2}+x^{2} = 1
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.686 |
|
|
\[
{}y^{\prime \prime } = \frac {1}{\sqrt {y}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
12.609 |
|
|
\[
{}a^{3} y^{\prime \prime \prime } y^{\prime \prime } = \sqrt {1+c^{2} {y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
1.427 |
|
|
\[
{}y^{\prime \prime \prime } = \sqrt {1+{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
3.898 |
|
|
\[
{}2 \left (2 a -y\right ) y^{\prime \prime } = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.597 |
|
|
\[
{}y^{\prime \prime }-x y^{\prime \prime \prime }+{y^{\prime \prime \prime }}^{3} = 0
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✓ |
0.582 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2} = y^{2} \ln \left (y\right )
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.316 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.320 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.148 |
|
|
\[
{}n \,x^{3} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.148 |
|