|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x \left (y^{2}-3 x \right ) y^{\prime }+2 y^{3}-5 x y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.567 |
|
|
\[
{}x \left (y^{2}+x^{2}-a \right ) y^{\prime }-\left (a +x^{2}+y^{2}\right ) y = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
7.570 |
|
|
\[
{}x \left (y^{2}+x y-x^{2}\right ) y^{\prime }-y^{3}+x y^{2}+x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
215.765 |
|
|
\[
{}x \left (y^{2}+x^{2} y+x^{2}\right ) y^{\prime }-2 y^{3}-2 x^{2} y^{2}+x^{4} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.227 |
|
|
\[
{}2 x \left (5 x^{2}+y^{2}\right ) y^{\prime }+y^{3}-x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
43.756 |
|
|
\[
{}3 x y^{2} y^{\prime }+y^{3}-2 x = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
2.324 |
|
|
\[
{}\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 x y = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
1.999 |
|
|
\[
{}6 x y^{2} y^{\prime }+x +2 y^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
2.701 |
|
|
\[
{}\left (x^{2}+6 x y^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.719 |
|
|
\[
{}\left (x^{2} y^{2}+x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
7.684 |
|
|
\[
{}\left (x y-1\right )^{2} x y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.963 |
|
|
\[
{}\left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.182 |
|
|
\[
{}\left (y^{3}-3 x \right ) y^{\prime }-3 y+x^{2} = 0
\] |
[_exact, _rational] |
✓ |
1.367 |
|
|
\[
{}\left (y^{3}-x^{3}\right ) y^{\prime }-x^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
143.296 |
|
|
\[
{}\left (a +x^{2}+y^{2}\right ) y y^{\prime }+\left (y^{2}+x^{2}-a \right ) x = 0
\] |
[_exact, _rational] |
✓ |
2.489 |
|
|
\[
{}2 y^{3} y^{\prime }+x y^{2} = 0
\] |
[_separable] |
✓ |
3.197 |
|
|
\[
{}\left (2 y^{3}+y\right ) y^{\prime }-2 x^{3}-x = 0
\] |
[_separable] |
✓ |
2.755 |
|
|
\[
{}\left (2 y^{3}+5 x^{2} y\right ) y^{\prime }+5 x y^{2}+x^{3} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
215.760 |
|
|
\[
{}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
252.637 |
|
|
\[
{}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b} = 0
\] |
[_rational] |
✗ |
4.272 |
|
|
\[
{}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0
\] |
[_rational] |
✗ |
3.668 |
|
|
\[
{}x y^{3} y^{\prime }+y^{4}-x \sin \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
36.580 |
|
|
\[
{}\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.503 |
|
|
\[
{}\left (2 x y^{3}+y\right ) y^{\prime }+2 y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.229 |
|
|
\[
{}\left (2 x y^{3}+x y+x^{2}\right ) y^{\prime }-x y+y^{2} = 0
\] |
[_rational] |
✓ |
1.720 |
|
|
\[
{}\left (3 x y^{3}-4 x y+y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right ) = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.745 |
|
|
\[
{}\left (7 x y^{3}+y-5 x \right ) y^{\prime }+y^{4}-5 y = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.862 |
|
|
\[
{}\left (x^{2} y^{3}+x y\right ) y^{\prime }-1 = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.658 |
|
|
\[
{}\left (2 x^{2} y^{3}+x^{2} y^{2}-2 x \right ) y^{\prime }-2 y-1 = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.745 |
|
|
\[
{}\left (10 x^{2} y^{3}-3 y^{2}-2\right ) y^{\prime }+5 x y^{4}+x = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.619 |
|
|
\[
{}\left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y = 0
\] |
[_rational] |
✓ |
1.960 |
|
|
\[
{}\left (2 x^{3} y^{3}-x \right ) y^{\prime }+2 x^{3} y^{3}-y = 0
\] |
[_rational] |
✓ |
1.678 |
|
|
\[
{}y \left (y^{3}-2 x^{3}\right ) y^{\prime }+\left (2 y^{3}-x^{3}\right ) x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
78.754 |
|
|
\[
{}y \left (\left (b x +a y\right )^{3}+b \,x^{3}\right ) y^{\prime }+x \left (\left (b x +a y\right )^{3}+a y^{3}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.214 |
|
|
\[
{}\left (x +2 y+2 x^{2} y^{3}+x y^{4}\right ) y^{\prime }+y^{5}+y = 0
\] |
[_rational] |
✓ |
3.659 |
|
|
\[
{}a \,x^{2} y^{n} y^{\prime }-2 y^{\prime } x +y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.743 |
|
|
\[
{}y^{m} x^{n} \left (a x y^{\prime }+b y\right )+\alpha x y^{\prime }+\beta y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.691 |
|
|
\[
{}\left (f \left (x +y\right )+1\right ) y^{\prime }+f \left (x +y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _dAlembert] |
✓ |
1.762 |
|
|
\[
{}\left (\sqrt {x y}-1\right ) x y^{\prime }-\left (\sqrt {x y}+1\right ) y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
5.473 |
|
|
\[
{}\left (2 x^{{5}/{2}} y^{{3}/{2}}+x^{2} y-x \right ) y^{\prime }-x^{{3}/{2}} y^{{5}/{2}}+x y^{2}-y = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
101.311 |
|
|
\[
{}\left (1+\sqrt {x +y}\right ) y^{\prime }+1 = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.768 |
|
|
\[
{}\sqrt {-1+y^{2}}\, y^{\prime }-\sqrt {x^{2}-1} = 0
\] |
[_separable] |
✓ |
2.440 |
|
|
\[
{}\left (\sqrt {1+y^{2}}+a x \right ) y^{\prime }+\sqrt {x^{2}+1}+a y = 0
\] |
[_exact] |
✓ |
36.473 |
|
|
\[
{}\left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
7.384 |
|
|
\[
{}\left (y \sqrt {x^{2}+y^{2}}+\left (y^{2}-x^{2}\right ) \sin \left (\alpha \right )-2 x y \cos \left (\alpha \right )\right ) y^{\prime }+x \sqrt {x^{2}+y^{2}}+2 x y \sin \left (\alpha \right )+\left (y^{2}-x^{2}\right ) \cos \left (\alpha \right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
185.713 |
|
|
\[
{}\left (x \sqrt {x^{2}+y^{2}+1}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime }-y \sqrt {x^{2}+y^{2}+1}-x \left (x^{2}+y^{2}\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.610 |
|
|
\[
{}\left (\frac {\operatorname {e1} \left (x +a \right )}{\left (\left (x +a \right )^{2}+y^{2}\right )^{{3}/{2}}}+\frac {\operatorname {e2} \left (x -a \right )}{\left (\left (x -a \right )^{2}+y^{2}\right )^{{3}/{2}}}\right ) y^{\prime }-y \left (\frac {\operatorname {e1}}{\left (\left (x +a \right )^{2}+y^{2}\right )^{{3}/{2}}}+\frac {\operatorname {e2}}{\left (\left (x -a \right )^{2}+y^{2}\right )^{{3}/{2}}}\right ) = 0
\] |
unknown |
✓ |
238.041 |
|
|
\[
{}\left ({\mathrm e}^{x}+x \,{\mathrm e}^{y}\right ) y^{\prime }+y \,{\mathrm e}^{x}+{\mathrm e}^{y} = 0
\] |
[_exact] |
✓ |
1.575 |
|
|
\[
{}x \left (3 \,{\mathrm e}^{x y}+2 \,{\mathrm e}^{-x y}\right ) \left (y^{\prime } x +y\right )+1 = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
31.299 |
|
|
\[
{}\left (\ln \left (y\right )+x \right ) y^{\prime }-1 = 0
\] |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
1.382 |
|
|
\[
{}\left (\ln \left (y\right )+2 x -1\right ) y^{\prime }-2 y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.815 |
|
|
\[
{}x \left (2 x^{2} y \ln \left (y\right )+1\right ) y^{\prime }-2 y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.751 |
|
|
\[
{}x \left (y \ln \left (x y\right )+y-a x \right ) y^{\prime }-y \left (a x \ln \left (x y\right )-y+a x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
2.132 |
|
|
\[
{}y^{\prime } \left (1+\sin \left (x \right )\right ) \sin \left (y\right )+\cos \left (x \right ) \left (\cos \left (y\right )-1\right ) = 0
\] |
[_separable] |
✓ |
5.546 |
|
|
\[
{}\sin \left (y\right )+\cos \left (x \right ) y+\left (x \cos \left (y\right )+\sin \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
9.001 |
|
|
\[
{}x y^{\prime } \cot \left (\frac {y}{x}\right )+2 x \sin \left (\frac {y}{x}\right )-y \cot \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.227 |
|
|
\[
{}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
49.634 |
|
|
\[
{}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
42.231 |
|
|
\[
{}y^{\prime } \left (\cos \left (y\right )-\sin \left (\alpha \right ) \sin \left (x \right )\right ) \cos \left (y\right )+\left (\cos \left (x \right )-\sin \left (\alpha \right ) \sin \left (y\right )\right ) \cos \left (x \right ) = 0
\] |
unknown |
✓ |
47.341 |
|
|
\[
{}x \cos \left (y\right ) y^{\prime }+\sin \left (y\right ) = 0
\] |
[_separable] |
✓ |
4.540 |
|
|
\[
{}\left (x \sin \left (y\right )-1\right ) y^{\prime }+\cos \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
5.320 |
|
|
\[
{}\left (x \cos \left (y\right )+\cos \left (x \right )\right ) y^{\prime }-y \sin \left (x \right )+\sin \left (y\right ) = 0
\] |
[_exact] |
✓ |
30.127 |
|
|
\[
{}\left (x^{2} \cos \left (y\right )+2 y \sin \left (x \right )\right ) y^{\prime }+2 x \sin \left (y\right )+y^{2} \cos \left (x \right ) = 0
\] |
[_exact] |
✓ |
40.236 |
|
|
\[
{}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
59.908 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )+\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.683 |
|
|
\[
{}3 \sin \left (x \right ) \sin \left (y\right ) y^{\prime }+5 \cos \left (x \right )^{4} y = 0
\] |
[_separable] |
✓ |
5.348 |
|
|
\[
{}y^{\prime } \cos \left (a y\right )-b \left (1-c \cos \left (a y\right )\right ) \sqrt {\cos \left (a y\right )^{2}-1+c \cos \left (a y\right )} = 0
\] |
[_quadrature] |
✓ |
21.756 |
|
|
\[
{}\left (x \sin \left (x y\right )+\cos \left (x +y\right )-\sin \left (y\right )\right ) y^{\prime }+y \sin \left (x y\right )+\cos \left (x +y\right )+\cos \left (x \right ) = 0
\] |
[_exact] |
✓ |
41.912 |
|
|
\[
{}\left (x^{2} y \sin \left (x y\right )-4 x \right ) y^{\prime }+x y^{2} \sin \left (x y\right )-y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
39.114 |
|
|
\[
{}\left (y^{\prime } x -y\right ) \cos \left (\frac {y}{x}\right )^{2}+x = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
10.625 |
|
|
\[
{}\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }-\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.345 |
|
|
\[
{}\left (y f \left (x^{2}+y^{2}\right )-x \right ) y^{\prime }+y+x f \left (x^{2}+y^{2}\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.573 |
|
|
\[
{}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-y^{\prime } x = 0
\] |
[_exact] |
✗ |
3.407 |
|
|
\[
{}{y^{\prime }}^{2}+a y+b \,x^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.117 |
|
|
\[
{}{y^{\prime }}^{2}+y^{2}-a^{2} = 0
\] |
[_quadrature] |
✓ |
1.171 |
|
|
\[
{}{y^{\prime }}^{2}-y^{3}+y^{2} = 0
\] |
[_quadrature] |
✓ |
20.208 |
|
|
\[
{}{y^{\prime }}^{2}-4 y^{3}+a y+b = 0
\] |
[_quadrature] |
✓ |
141.202 |
|
|
\[
{}{y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right ) = 0
\] |
[_quadrature] |
✓ |
0.916 |
|
|
\[
{}{y^{\prime }}^{2}-2 y^{\prime }-y^{2} = 0
\] |
[_quadrature] |
✓ |
39.059 |
|
|
\[
{}{y^{\prime }}^{2}+a y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.341 |
|
|
\[
{}{y^{\prime }}^{2}+a y^{\prime }+b y = 0
\] |
[_quadrature] |
✓ |
0.702 |
|
|
\[
{}{y^{\prime }}^{2}+\left (x -2\right ) y^{\prime }-y+1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.567 |
|
|
\[
{}{y^{\prime }}^{2}+\left (x +a \right ) y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.591 |
|
|
\[
{}{y^{\prime }}^{2}-\left (x +1\right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.552 |
|
|
\[
{}{y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.543 |
|
|
\[
{}{y^{\prime }}^{2}-2 y^{\prime } x +y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.554 |
|
|
\[
{}{y^{\prime }}^{2}+a x y^{\prime }-b \,x^{2}-c = 0
\] |
[_quadrature] |
✓ |
0.664 |
|
|
\[
{}{y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
3.016 |
|
|
\[
{}{y^{\prime }}^{2}+\left (a x +b \right ) y^{\prime }-a y+c = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.665 |
|
|
\[
{}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.070 |
|
|
\[
{}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.502 |
|
|
\[
{}{y^{\prime }}^{2}+\left (y^{\prime }-y\right ) {\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.828 |
|
|
\[
{}{y^{\prime }}^{2}-2 y y^{\prime }-2 x = 0
\] |
[_dAlembert] |
✓ |
82.163 |
|
|
\[
{}{y^{\prime }}^{2}-\left (4 y+1\right ) y^{\prime }+\left (4 y+1\right ) y = 0
\] |
[_quadrature] |
✓ |
1.924 |
|
|
\[
{}{y^{\prime }}^{2}+a y y^{\prime }-b x -c = 0
\] |
[_dAlembert] |
✓ |
35.596 |
|
|
\[
{}{y^{\prime }}^{2}+\left (b x +a y\right ) y^{\prime }+a b x y = 0
\] |
[_quadrature] |
✓ |
0.470 |
|
|
\[
{}{y^{\prime }}^{2}-x y^{\prime } y+y^{2} \ln \left (a y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
5.361 |
|
|
\[
{}{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2} = 0
\] |
[_separable] |
✓ |
1.421 |
|
|
\[
{}{y^{\prime }}^{2}+y \left (y-x \right ) y^{\prime }-x y^{3} = 0
\] |
[_separable] |
✓ |
0.501 |
|
|
\[
{}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.451 |
|