|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\sqrt {x^{2}-1}\, y^{\prime }-\sqrt {y^{2}-1} = 0
\] |
[_separable] |
✓ |
14.502 |
|
|
\[
{}\sqrt {-x^{2}+1}\, y^{\prime }-y \sqrt {y^{2}-1} = 0
\] |
[_separable] |
✓ |
18.696 |
|
|
\[
{}\sqrt {a^{2}+x^{2}}\, y^{\prime }+y-\sqrt {a^{2}+x^{2}}+x = 0
\] |
[_linear] |
✓ |
1.553 |
|
|
\[
{}x y^{\prime } \ln \left (x \right )+y-a x \left (\ln \left (x \right )+1\right ) = 0
\] |
[_linear] |
✓ |
1.525 |
|
|
\[
{}x y^{\prime } \ln \left (x \right )-y^{2} \ln \left (x \right )-\left (2 \ln \left (x \right )^{2}+1\right ) y-\ln \left (x \right )^{3} = 0
\] |
[_Riccati] |
✓ |
2.499 |
|
|
\[
{}\sin \left (x \right ) y^{\prime }-y^{2} \sin \left (x \right )^{2}+\left (\cos \left (x \right )-3 \sin \left (x \right )\right ) y+4 = 0
\] |
[_Riccati] |
✓ |
9.587 |
|
|
\[
{}\cos \left (x \right ) y^{\prime }+y+\left (\sin \left (x \right )+1\right ) \cos \left (x \right ) = 0
\] |
[_linear] |
✓ |
3.166 |
|
|
\[
{}\cos \left (x \right ) y^{\prime }-y^{4}-y \sin \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
7.034 |
|
|
\[
{}\sin \left (x \right ) \cos \left (x \right ) y^{\prime }-y-\sin \left (x \right )^{3} = 0
\] |
[_linear] |
✓ |
4.322 |
|
|
\[
{}\sin \left (2 x \right ) y^{\prime }+\sin \left (2 y\right ) = 0
\] |
[_separable] |
✓ |
25.406 |
|
|
\[
{}\left (a \sin \left (x \right )^{2}+b \right ) y^{\prime }+a y \sin \left (2 x \right )+A x \left (a \sin \left (x \right )^{2}+c \right ) = 0
\] |
[_linear] |
✓ |
13.710 |
|
|
\[
{}2 f \left (x \right ) y^{\prime }+2 f \left (x \right ) y^{2}-f^{\prime }\left (x \right ) y-2 f \left (x \right )^{2} = 0
\] |
[_Riccati] |
✓ |
0.814 |
|
|
\[
{}y y^{\prime }+y+x^{3} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
0.706 |
|
|
\[
{}y y^{\prime }+a y+x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
11.483 |
|
|
\[
{}y y^{\prime }+a y+\frac {\left (a^{2}-1\right ) x}{4}+b \,x^{n} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
2.266 |
|
|
\[
{}y y^{\prime }+a y+b \,{\mathrm e}^{x}-2 a = 0
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
1.698 |
|
|
\[
{}y y^{\prime }+y^{2}+4 x \left (x +1\right ) = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.327 |
|
|
\[
{}y y^{\prime }+a y^{2}-b \cos \left (x +c \right ) = 0
\] |
[_Bernoulli] |
✓ |
3.311 |
|
|
\[
{}y y^{\prime }-\sqrt {a y^{2}+b} = 0
\] |
[_quadrature] |
✓ |
5.797 |
|
|
\[
{}y y^{\prime }+x y^{2}-4 x = 0
\] |
[_separable] |
✓ |
2.140 |
|
|
\[
{}y y^{\prime }-x \,{\mathrm e}^{\frac {x}{y}} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.238 |
|
|
\[
{}y y^{\prime }+f \left (x^{2}+y^{2}\right ) g \left (x \right )+x = 0
\] |
[NONE] |
✗ |
3.109 |
|
|
\[
{}\left (y+1\right ) y^{\prime }-y-x = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
6.415 |
|
|
\[
{}\left (y+x -1\right ) y^{\prime }-y+2 x +3 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.891 |
|
|
\[
{}\left (y+2 x -2\right ) y^{\prime }-y+x +1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.210 |
|
|
\[
{}\left (y-2 x +1\right ) y^{\prime }+y+x = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.272 |
|
|
\[
{}\left (y-x^{2}\right ) y^{\prime }-x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
1.072 |
|
|
\[
{}\left (y-x^{2}\right ) y^{\prime }+4 x y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.357 |
|
|
\[
{}2 y y^{\prime }-x y^{2}-x^{3} = 0
\] |
[_rational, _Bernoulli] |
✓ |
1.634 |
|
|
\[
{}\left (2 y+x +1\right ) y^{\prime }-2 y-x +1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.890 |
|
|
\[
{}\left (2 y+x +7\right ) y^{\prime }-y+2 x +4 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
36.780 |
|
|
\[
{}\left (2 y-x \right ) y^{\prime }-y-2 x = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.273 |
|
|
\[
{}\left (2 y-6 x \right ) y^{\prime }-y+3 x +2 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.013 |
|
|
\[
{}\left (4 y+2 x +3\right ) y^{\prime }-2 y-x -1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.885 |
|
|
\[
{}\left (4 y-2 x -3\right ) y^{\prime }+2 y-x -1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.891 |
|
|
\[
{}\left (4 y-3 x -5\right ) y^{\prime }-3 y+7 x +2 = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.442 |
|
|
\[
{}\left (4 y+11 x -11\right ) y^{\prime }-25 y-8 x +62 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.945 |
|
|
\[
{}\left (12 y-5 x -8\right ) y^{\prime }-5 y+2 x +3 = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.736 |
|
|
\[
{}a y y^{\prime }+b y^{2}+f \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.099 |
|
|
\[
{}\left (a y+b x +c \right ) y^{\prime }+\alpha y+\beta x +\gamma = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
8.820 |
|
|
\[
{}x y y^{\prime }+y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
75.413 |
|
|
\[
{}x y y^{\prime }-y^{2}+a \,x^{3} \cos \left (x \right ) = 0
\] |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
3.678 |
|
|
\[
{}x y y^{\prime }-y^{2}+x y+x^{3}-2 x^{2} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
1.091 |
|
|
\[
{}\left (x y+a \right ) y^{\prime }+b y = 0
\] |
[[_1st_order, _with_exponential_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.994 |
|
|
\[
{}x \left (y+4\right ) y^{\prime }-y^{2}-2 y-2 x = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.270 |
|
|
\[
{}x \left (y+a \right ) y^{\prime }+b y+c x = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
0.944 |
|
|
\[
{}\left (\left (x +y\right ) x +a \right ) y^{\prime }-y \left (x +y\right )-b = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.469 |
|
|
\[
{}\left (x y-x^{2}\right ) y^{\prime }+y^{2}-3 x y-2 x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
77.637 |
|
|
\[
{}2 x y y^{\prime }-y^{2}+a x = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.752 |
|
|
\[
{}2 x y y^{\prime }-y^{2}+a \,x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
4.000 |
|
|
\[
{}2 x y y^{\prime }+2 y^{2}+1 = 0
\] |
[_separable] |
✓ |
3.079 |
|
|
\[
{}x \left (2 y+x -1\right ) y^{\prime }-y \left (y+2 x +1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.483 |
|
|
\[
{}x \left (2 y-x -1\right ) y^{\prime }+y \left (2 x -y-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.508 |
|
|
\[
{}\left (2 x y+4 x^{3}\right ) y^{\prime }+y^{2}+112 x^{2} y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.093 |
|
|
\[
{}x \left (3 y+2 x \right ) y^{\prime }+3 \left (x +y\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
40.779 |
|
|
\[
{}\left (3 x +2\right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+x y-7 x^{2}-9 x -3 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.437 |
|
|
\[
{}\left (6 x y+x^{2}+3\right ) y^{\prime }+3 y^{2}+2 x y+2 x = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.557 |
|
|
\[
{}\left (y a x +b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
6.047 |
|
|
\[
{}\left (B x y+A \,x^{2}+a x +b y+c \right ) y^{\prime }-B g \left (x \right )^{2}+A x y+\alpha x +\beta y+\gamma = 0
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
8.679 |
|
|
\[
{}\left (x^{2} y-1\right ) y^{\prime }+x y^{2}-1 = 0
\] |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.398 |
|
|
\[
{}\left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1 = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
1.930 |
|
|
\[
{}\left (x^{2} y-1\right ) y^{\prime }+8 x y^{2}-8 = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
1.022 |
|
|
\[
{}x \left (x y-2\right ) y^{\prime }+x^{2} y^{3}+x y^{2}-2 y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.652 |
|
|
\[
{}x \left (x y-3\right ) y^{\prime }+x y^{2}-y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.267 |
|
|
\[
{}x^{2} \left (y-1\right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[_separable] |
✓ |
1.579 |
|
|
\[
{}x \left (x y+x^{4}-1\right ) y^{\prime }-y \left (x y-x^{4}-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
1.370 |
|
|
\[
{}2 x^{2} y y^{\prime }+y^{2}-2 x^{3}-x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.479 |
|
|
\[
{}2 x^{2} y y^{\prime }-y^{2}-x^{2} {\mathrm e}^{x -\frac {1}{x}} = 0
\] |
[_Bernoulli] |
✓ |
2.557 |
|
|
\[
{}\left (2 x^{2} y+x \right ) y^{\prime }-x^{2} y^{3}+2 x y^{2}+y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
2.748 |
|
|
\[
{}\left (2 x^{2} y-x \right ) y^{\prime }-2 x y^{2}-y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.989 |
|
|
\[
{}\left (2 x^{2} y-x^{3}\right ) y^{\prime }+y^{3}-4 x y^{2}+2 x^{3} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
36.318 |
|
|
\[
{}2 x^{3}+y y^{\prime }+3 y^{2} x^{2}+7 = 0
\] |
[_rational, _Bernoulli] |
✓ |
2.610 |
|
|
\[
{}2 x \left (x^{3} y+1\right ) y^{\prime }+\left (3 x^{3} y-1\right ) y = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
2.211 |
|
|
\[
{}\left (x^{\left (n +1\right ) n} y-1\right ) y^{\prime }+2 \left (n +1\right )^{2} x^{n -1} \left (x^{n^{2}} y^{2}-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
5.994 |
|
|
\[
{}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (1+y^{2}\right )^{3}} = 0
\] |
[‘x=_G(y,y’)‘] |
✓ |
128.947 |
|
|
\[
{}y y^{\prime } \sin \left (x \right )^{2}+y^{2} \cos \left (x \right ) \sin \left (x \right )-1 = 0
\] |
[_exact, _Bernoulli] |
✓ |
6.432 |
|
|
\[
{}f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}+h \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
1.788 |
|
|
\[
{}\left (y^{2}-x \right ) y^{\prime }-y+x^{2} = 0
\] |
[_exact, _rational] |
✓ |
1.178 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right ) = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.226 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime }-y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
3668.745 |
|
|
\[
{}\left (y^{2}+x^{2}+a \right ) y^{\prime }+2 x y = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.237 |
|
|
\[
{}\left (y^{2}+x^{2}+a \right ) y^{\prime }+2 x y+x^{2}+b = 0
\] |
[_exact, _rational] |
✓ |
1.309 |
|
|
\[
{}\left (y^{2}+x^{2}+x \right ) y^{\prime }-y = 0
\] |
[_rational] |
✓ |
1.377 |
|
|
\[
{}\left (y^{2}-x^{2}\right ) y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.646 |
|
|
\[
{}\left (y^{2}+x^{4}\right ) y^{\prime }-4 x^{3} y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.903 |
|
|
\[
{}\left (y^{2}+4 \sin \left (x \right )\right ) y^{\prime }-\cos \left (x \right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.055 |
|
|
\[
{}\left (y^{2}+2 y+x \right ) y^{\prime }+\left (x +y\right )^{2} y^{2}+y \left (y+1\right ) = 0
\] |
[_rational] |
✗ |
4.078 |
|
|
\[
{}\left (x +y\right )^{2} y^{\prime }-a^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
4.965 |
|
|
\[
{}\left (y^{2}+2 x y-x^{2}\right ) y^{\prime }-y^{2}+2 x y+x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
33.139 |
|
|
\[
{}\left (y+3 x -1\right )^{2} y^{\prime }-\left (2 y-1\right ) \left (4 y+6 x -3\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
3.010 |
|
|
\[
{}3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (x +1\right ) y-3 \,{\mathrm e}^{x} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.972 |
|
|
\[
{}\left (4 y^{2}+x^{2}\right ) y^{\prime }-x y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
15.624 |
|
|
\[
{}\left (4 y^{2}+2 x y+3 x^{2}\right ) y^{\prime }+y^{2}+6 x y+2 x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.985 |
|
|
\[
{}\left (2 y-3 x +1\right )^{2} y^{\prime }-\left (3 y-2 x -4\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
36.094 |
|
|
\[
{}\left (2 y-4 x +1\right )^{2} y^{\prime }-\left (y-2 x \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
2.211 |
|
|
\[
{}\left (6 y^{2}-3 x^{2} y+1\right ) y^{\prime }-3 x y^{2}+x = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.408 |
|
|
\[
{}\left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 x y+a = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.454 |
|
|
\[
{}\left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
312.763 |
|
|
\[
{}\left (b \left (\beta y+\alpha x \right )^{2}-\beta \left (a x +b y\right )\right ) y^{\prime }+a \left (\beta y+\alpha x \right )^{2}-\alpha \left (a x +b y\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
2.138 |
|
|
\[
{}\left (a y+b x +c \right )^{2} y^{\prime }+\left (\alpha y+\beta x +\gamma \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
2.733 |
|