|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\frac {x y}{\sqrt {x^{2}+1}}+2 x y-\frac {y}{x}+\left (\sqrt {x^{2}+1}+x^{2}-\ln \left (x \right )\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
18.625 |
|
|
\[
{}\sin \left (y\right )+y \sin \left (x \right )+\frac {1}{x}+\left (x \cos \left (y\right )-\cos \left (x \right )+\frac {1}{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
35.850 |
|
|
\[
{}\frac {y+\sin \left (x \right ) \cos \left (x y\right )^{2}}{\cos \left (x y\right )^{2}}+\left (\frac {x}{\cos \left (x y\right )^{2}}+\sin \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
51.612 |
|
|
\[
{}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
10.831 |
|
|
\[
{}y \left (x^{2}+y^{2}+a^{2}\right ) y^{\prime }+x \left (x^{2}+y^{2}-a^{2}\right ) = 0
\] |
[_exact, _rational] |
✓ |
1.684 |
|
|
\[
{}3 x^{2} y+y^{3}+\left (x^{3}+3 x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
109.938 |
|
|
\[
{}1-x^{2} y+x^{2} \left (y-x \right ) y^{\prime } = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.244 |
|
|
\[
{}x^{2}+y-y^{\prime } x = 0
\] |
[_linear] |
✓ |
1.525 |
|
|
\[
{}x +y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.792 |
|
|
\[
{}2 x^{2} y+2 y+5+\left (2 x^{3}+2 x \right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
1.336 |
|
|
\[
{}x^{4} \ln \left (x \right )-2 x y^{3}+3 x^{2} y^{2} y^{\prime } = 0
\] |
[_Bernoulli] |
✓ |
1.712 |
|
|
\[
{}x +\sin \left (x \right )+\sin \left (y\right )+y^{\prime } \cos \left (y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
3.591 |
|
|
\[
{}2 x y^{2}-3 y^{3}+\left (7-3 x y^{2}\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
2.345 |
|
|
\[
{}3 y^{2}-x +\left (2 y^{3}-6 x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.984 |
|
|
\[
{}x^{2}+y^{2}+1-2 x y y^{\prime } = 0
\] |
[_rational, _Bernoulli] |
✓ |
2.058 |
|
|
\[
{}x -x y+\left (y+x^{2}\right ) y^{\prime } = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.189 |
|
|
\[
{}4 {y^{\prime }}^{2}-9 x = 0
\] |
[_quadrature] |
✓ |
0.250 |
|
|
\[
{}{y^{\prime }}^{2}-2 y y^{\prime } = y^{2} \left ({\mathrm e}^{2 x}-1\right )
\] |
[_separable] |
✓ |
0.419 |
|
|
\[
{}{y^{\prime }}^{2}-2 y^{\prime } x -8 x^{2} = 0
\] |
[_quadrature] |
✓ |
0.875 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+2 y^{2} = 0
\] |
[_separable] |
✓ |
4.115 |
|
|
\[
{}{y^{\prime }}^{2}-\left (2 x +y\right ) y^{\prime }+x^{2}+x y = 0
\] |
[_quadrature] |
✓ |
1.575 |
|
|
\[
{}{y^{\prime }}^{3}+\left (x +2\right ) {\mathrm e}^{y} = 0
\] |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
0.766 |
|
|
\[
{}{y^{\prime }}^{3} = y {y^{\prime }}^{2}-x^{2} y^{\prime }+x^{2} y
\] |
[_quadrature] |
✓ |
1.559 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.200 |
|
|
\[
{}{y^{\prime }}^{2}-4 y^{\prime } x +2 y+2 x^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.114 |
|
|
\[
{}y = {y^{\prime }}^{2} {\mathrm e}^{y^{\prime }}
\] |
[_quadrature] |
✓ |
1.556 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\frac {y^{\prime }}{y}}
\] |
[_quadrature] |
✓ |
2.970 |
|
|
\[
{}x = \ln \left (y^{\prime }\right )+\sin \left (y^{\prime }\right )
\] |
[_quadrature] |
✓ |
1.891 |
|
|
\[
{}x = {y^{\prime }}^{2}-2 y^{\prime }+2
\] |
[_quadrature] |
✓ |
0.201 |
|
|
\[
{}y = y^{\prime } \ln \left (y^{\prime }\right )
\] |
[_quadrature] |
✓ |
4.282 |
|
|
\[
{}y = \left (y^{\prime }-1\right ) {\mathrm e}^{y^{\prime }}
\] |
[_quadrature] |
✓ |
1.309 |
|
|
\[
{}{y^{\prime }}^{2} x = {\mathrm e}^{\frac {1}{y^{\prime }}}
\] |
[_quadrature] |
✓ |
0.491 |
|
|
\[
{}x \left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} = a
\] |
[_quadrature] |
✓ |
1.878 |
|
|
\[
{}y^{{2}/{5}}+{y^{\prime }}^{{2}/{5}} = a^{{2}/{5}}
\] |
[_quadrature] |
✓ |
2.355 |
|
|
\[
{}x = y^{\prime }+\sin \left (y^{\prime }\right )
\] |
[_quadrature] |
✓ |
0.503 |
|
|
\[
{}y = y^{\prime } \left (1+y^{\prime } \cos \left (y^{\prime }\right )\right )
\] |
[_quadrature] |
✓ |
1.663 |
|
|
\[
{}y = \arcsin \left (y^{\prime }\right )+\ln \left ({y^{\prime }}^{2}+1\right )
\] |
[_quadrature] |
✓ |
3.309 |
|
|
\[
{}y = 2 y^{\prime } x +\ln \left (y^{\prime }\right )
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
2.329 |
|
|
\[
{}y = x \left (1+y^{\prime }\right )+{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.460 |
|
|
\[
{}y = 2 y^{\prime } x +\sin \left (y^{\prime }\right )
\] |
[_dAlembert] |
✓ |
1.117 |
|
|
\[
{}y = {y^{\prime }}^{2} x -\frac {1}{y^{\prime }}
\] |
[_dAlembert] |
✓ |
2.914 |
|
|
\[
{}y = \frac {3 y^{\prime } x}{2}+{\mathrm e}^{y^{\prime }}
\] |
[_dAlembert] |
✓ |
1.020 |
|
|
\[
{}y = y^{\prime } x +\frac {a}{{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.760 |
|
|
\[
{}y = y^{\prime } x +{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.318 |
|
|
\[
{}{y^{\prime }}^{2} x -y y^{\prime }-y^{\prime }+1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.465 |
|
|
\[
{}y = y^{\prime } x +a \sqrt {{y^{\prime }}^{2}+1}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
6.006 |
|
|
\[
{}x = \frac {y}{y^{\prime }}+\frac {1}{{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
0.352 |
|
|
\[
{}y^{\prime } {\mathrm e}^{-x}+y^{2}-2 y \,{\mathrm e}^{x} = 1-{\mathrm e}^{2 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.210 |
|
|
\[
{}y^{\prime }+y^{2}-2 y \sin \left (x \right )+\sin \left (x \right )^{2}-\cos \left (x \right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.524 |
|
|
\[
{}y^{\prime } x -y^{2}+\left (2 x +1\right ) y = x^{2}+2 x
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
1.808 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+x y+1
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.608 |
|
|
\[
{}\left ({y^{\prime }}^{2}+1\right ) y^{2}-4 y y^{\prime }-4 x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
6.946 |
|
|
\[
{}{y^{\prime }}^{2}-4 y = 0
\] |
[_quadrature] |
✓ |
0.536 |
|
|
\[
{}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
7.220 |
|
|
\[
{}{y^{\prime }}^{2}-y^{2} = 0
\] |
[_quadrature] |
✓ |
2.439 |
|
|
\[
{}y^{\prime } = y^{{2}/{3}}+a
\] |
[_quadrature] |
✓ |
3.003 |
|
|
\[
{}\left (y^{\prime } x +y\right )^{2}+3 x^{5} \left (y^{\prime } x -2 y\right ) = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
8.421 |
|
|
\[
{}y \left (y-2 y^{\prime } x \right )^{2} = 2 y^{\prime }
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.059 |
|
|
\[
{}8 {y^{\prime }}^{3}-12 {y^{\prime }}^{2} = 27 y-27 x
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.677 |
|
|
\[
{}\left (y^{\prime }-1\right )^{2} = y^{2}
\] |
[_quadrature] |
✓ |
2.039 |
|
|
\[
{}y = {y^{\prime }}^{2}-y^{\prime } x +x
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.467 |
|
|
\[
{}\left (y^{\prime } x +y\right )^{2} = y^{2} y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.600 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+y^{2} = 1
\] |
[_quadrature] |
✓ |
0.532 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.176 |
|
|
\[
{}3 {y^{\prime }}^{2} x -6 y y^{\prime }+x +2 y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.308 |
|
|
\[
{}y = y^{\prime } x +\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
7.245 |
|
|
\[
{}y^{\prime } = \left (x -y\right )^{2}+1
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
2.684 |
|
|
\[
{}x \sin \left (x \right ) y^{\prime }+\left (\sin \left (x \right )-x \cos \left (x \right )\right ) y = \sin \left (x \right ) \cos \left (x \right )-x
\] |
[_linear] |
✓ |
6.703 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = y^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
4.940 |
|
|
\[
{}x^{3}-3 x y^{2}+\left (y^{3}-3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
14.058 |
|
|
\[
{}5 x y-4 y^{2}-6 x^{2}+\left (y^{2}-8 x y+\frac {5 x^{2}}{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
129.046 |
|
|
\[
{}3 x y^{2}-x^{2}+\left (3 x^{2} y-6 y^{2}-1\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.504 |
|
|
\[
{}y-x y^{2} \ln \left (x \right )+y^{\prime } x = 0
\] |
[_Bernoulli] |
✓ |
2.284 |
|
|
\[
{}2 x y \,{\mathrm e}^{x^{2}}-x \sin \left (x \right )+{\mathrm e}^{x^{2}} y^{\prime } = 0
\] |
[_linear] |
✓ |
2.326 |
|
|
\[
{}y^{\prime } = \frac {1}{2 x -y^{2}}
\] |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
1.101 |
|
|
\[
{}x^{2}+y^{\prime } x = 3 x +y^{\prime }
\] |
[_quadrature] |
✓ |
0.555 |
|
|
\[
{}x y y^{\prime }-y^{2} = x^{4}
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
3.312 |
|
|
\[
{}\frac {1}{y^{2}-x y+x^{2}} = \frac {y^{\prime }}{2 y^{2}-x y}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
25.438 |
|
|
\[
{}\left (2 x -1\right ) y^{\prime }-2 y = \frac {1-4 x}{x^{2}}
\] |
[_linear] |
✓ |
1.266 |
|
|
\[
{}x -y+3+\left (3 x +y+1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.448 |
|
|
\[
{}y^{\prime }+\cos \left (\frac {x}{2}+\frac {y}{2}\right ) = \cos \left (\frac {x}{2}-\frac {y}{2}\right )
\] |
[_separable] |
✓ |
5.408 |
|
|
\[
{}y^{\prime } \left (3 x^{2}-2 x \right )-y \left (6 x -2\right ) = 0
\] |
[_separable] |
✓ |
1.733 |
|
|
\[
{}x y^{2} y^{\prime }-y^{3} = \frac {x^{4}}{3}
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.985 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
5.123 |
|
|
\[
{}x^{2}+y^{2}-x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
5.356 |
|
|
\[
{}x -y+2+\left (x -y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.762 |
|
|
\[
{}x y^{2}+y-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.143 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.114 |
|
|
\[
{}\left (x -1\right ) \left (y^{2}-y+1\right ) = \left (-1+y\right ) \left (x^{2}+x +1\right ) y^{\prime }
\] |
[_separable] |
✓ |
1.930 |
|
|
\[
{}\left (x -2 x y-y^{2}\right ) y^{\prime }+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.877 |
|
|
\[
{}y \cos \left (x \right )+\left (2 y-\sin \left (x \right )\right ) y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.036 |
|
|
\[
{}y^{\prime }-1 = {\mathrm e}^{x +2 y}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.263 |
|
|
\[
{}2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.164 |
|
|
\[
{}x^{2} y^{n} y^{\prime } = 2 y^{\prime } x -y
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.299 |
|
|
\[
{}\left (3 x +3 y+a^{2}\right ) y^{\prime } = 4 x +4 y+b^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.807 |
|
|
\[
{}x -y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.875 |
|
|
\[
{}y^{\prime } x +y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.419 |
|
|
\[
{}\sin \left (\ln \left (x \right )\right )-\cos \left (\ln \left (y\right )\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.176 |
|
|
\[
{}y^{\prime } = \sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.198 |
|
|
\[
{}\left (5 x -7 y+1\right ) y^{\prime }+x +y-1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.659 |
|