|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
0.319 |
|
|
\[
{}2 y x +\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.302 |
|
|
\[
{}y+\left (2 y-x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.307 |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-2 a}
\] |
[_rational, _Riccati] |
✓ |
0.537 |
|
|
\[
{}y^{\prime } x -a y+y^{2} = x^{-\frac {2 a}{3}}
\] |
[_rational, _Riccati] |
✓ |
1.744 |
|
|
\[
{}u^{\prime }+u^{2} = \frac {c}{x^{{4}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.298 |
|
|
\[
{}u^{\prime }+b u^{2} = \frac {c}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.274 |
|
|
\[
{}u^{\prime }-u^{2} = \frac {2}{x^{{8}/{3}}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.386 |
|
|
\[
{}\frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}} = -1
\] |
[_separable] |
✓ |
10.602 |
|
|
\[
{}{y^{\prime }}^{2}-5 y^{\prime }+6 = 0
\] |
[_quadrature] |
✓ |
0.724 |
|
|
\[
{}{y^{\prime }}^{2}-\frac {a^{2}}{x^{2}} = 0
\] |
[_quadrature] |
✓ |
0.346 |
|
|
\[
{}{y^{\prime }}^{2} = \frac {1-x}{x}
\] |
[_quadrature] |
✓ |
0.256 |
|
|
\[
{}{y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1 = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.018 |
|
|
\[
{}y = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.567 |
|
|
\[
{}x = a y^{\prime }+b {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
0.196 |
|
|
\[
{}y = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.573 |
|
|
\[
{}x = \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }
\] |
[_quadrature] |
✓ |
1.033 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1+{y^{\prime }}^{2}}}{x} = 0
\] |
[_quadrature] |
✓ |
1.095 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
1.993 |
|
|
\[
{}1+{y^{\prime }}^{2} = \frac {\left (x +a \right )^{2}}{2 a x +x^{2}}
\] |
[_quadrature] |
✓ |
0.562 |
|
|
\[
{}y = y^{\prime } x +y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.335 |
|
|
\[
{}y = y^{\prime } x +\sqrt {b^{2}-a^{2} {y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
3.244 |
|
|
\[
{}y = y^{\prime } x +x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
34.504 |
|
|
\[
{}y = y^{\prime } x +a x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✗ |
350.635 |
|
|
\[
{}x -y y^{\prime } = a {y^{\prime }}^{2}
\] |
unknown |
✓ |
428.598 |
|
|
\[
{}x +y y^{\prime } = a \sqrt {1+{y^{\prime }}^{2}}
\] |
unknown |
✗ |
477.262 |
|
|
\[
{}y y^{\prime } = x +y^{2}-y^{2} {y^{\prime }}^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
4.875 |
|
|
\[
{}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = x +\frac {y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.536 |
|
|
\[
{}y-2 y^{\prime } x = x {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.145 |
|
|
\[
{}\frac {y-y^{\prime } x}{y^{2}+y^{\prime }} = \frac {y-y^{\prime } x}{1+x^{2} y^{\prime }}
\] |
[_separable] |
✓ |
0.492 |
|
|
\[
{}2 y x +\left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
3.458 |
|
|
\[
{}\left (x +\sqrt {y^{2}-y x}\right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
4.767 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.451 |
|
|
\[
{}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.008 |
|
|
\[
{}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
24.788 |
|
|
\[
{}y^{2}+\left (x \sqrt {y^{2}-x^{2}}-y x \right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
10.177 |
|
|
\[
{}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.919 |
|
|
\[
{}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.496 |
|
|
\[
{}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.267 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.119 |
|
|
\[
{}x^{2}+y^{2} = 2 x y y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
9.752 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}+y = y^{\prime } x
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.754 |
|
|
\[
{}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.331 |
|
|
\[
{}y x -y^{2}-x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.686 |
|
|
\[
{}x +2 y-4-\left (2 x -4 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.754 |
|
|
\[
{}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.310 |
|
|
\[
{}x +y+1+\left (2 x +2 y+2\right ) y^{\prime } = 0
\] |
[_quadrature] |
✓ |
0.451 |
|
|
\[
{}x +y-1+\left (2 x +2 y-3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.206 |
|
|
\[
{}x +y-1-\left (x -y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.736 |
|
|
\[
{}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.167 |
|
|
\[
{}7 y-3+\left (2 x +1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.501 |
|
|
\[
{}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.269 |
|
|
\[
{}x +2 y+\left (-1+y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.616 |
|
|
\[
{}3 x -2 y+4-\left (2 x +7 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
46.129 |
|
|
\[
{}x +y+\left (3 x +3 y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.495 |
|
|
\[
{}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.847 |
|
|
\[
{}y+7+\left (2 x +y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
19.178 |
|
|
\[
{}x +y+2-\left (x -y-4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.750 |
|
|
\[
{}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
0.267 |
|
|
\[
{}\frac {2 y x +1}{y}+\frac {\left (-x +y\right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.319 |
|
|
\[
{}2 y x +\left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
0.211 |
|
|
\[
{}{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.270 |
|
|
\[
{}\cos \left (y\right )-\left (x \sin \left (y\right )-y^{2}\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
0.227 |
|
|
\[
{}x -2 y x +{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.640 |
|
|
\[
{}x^{2}-x +y^{2}-\left ({\mathrm e}^{y}-2 y x \right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.418 |
|
|
\[
{}2 x +y \cos \left (x \right )+\left (2 y+\sin \left (x \right )-\sin \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.253 |
|
|
\[
{}x \sqrt {x^{2}+y^{2}}-\frac {x^{2} y y^{\prime }}{y-\sqrt {x^{2}+y^{2}}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
0.274 |
|
|
\[
{}4 x^{3}-\sin \left (x \right )+y^{3}-\left (y^{2}+1-3 x y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.389 |
|
|
\[
{}{\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime } = 0
\] |
[_exact, _Bernoulli] |
✓ |
0.410 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )+\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
0.329 |
|
|
\[
{}y^{2} {\mathrm e}^{x y^{2}}+4 x^{3}+\left (2 x y \,{\mathrm e}^{x y^{2}}-3 y^{2}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.319 |
|
|
\[
{}y^{2}+y-y^{\prime } x = 0
\] |
[_separable] |
✓ |
0.190 |
|
|
\[
{}y \sec \left (x \right )+\sin \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
0.242 |
|
|
\[
{}{\mathrm e}^{x}-\sin \left (y\right )+\cos \left (y\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
0.254 |
|
|
\[
{}y x +\left (x^{2}+1\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
0.263 |
|
|
\[
{}y^{3}+x y^{2}+y+\left (x^{3}+x^{2} y+x \right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
0.526 |
|
|
\[
{}3 y-y^{\prime } x = 0
\] |
[_separable] |
✓ |
0.174 |
|
|
\[
{}y-3 y^{\prime } x = 0
\] |
[_separable] |
✓ |
0.208 |
|
|
\[
{}y \left (2 x^{2} y^{3}+3\right )+x \left (x^{2} y^{3}-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.786 |
|
|
\[
{}2 y x +x^{2}+\left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
0.228 |
|
|
\[
{}x^{2}+y \cos \left (x \right )+\left (y^{3}+\sin \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.223 |
|
|
\[
{}x^{2}+y^{2}+x +x y y^{\prime } = 0
\] |
[_rational, _Bernoulli] |
✓ |
0.310 |
|
|
\[
{}x -2 y x +{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.243 |
|
|
\[
{}{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
0.282 |
|
|
\[
{}x^{2}-y^{2}-y-\left (x^{2}-y^{2}-x \right ) y^{\prime } = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
1.421 |
|
|
\[
{}x^{4} y^{2}-y+\left (x^{2} y^{4}-x \right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
0.296 |
|
|
\[
{}y \left (2 x +y^{3}\right )-x \left (2 x -y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
0.280 |
|
|
\[
{}\arctan \left (y x \right )+\frac {y x -2 x y^{2}}{1+x^{2} y^{2}}+\frac {\left (x^{2}-2 x^{2} y\right ) y^{\prime }}{1+x^{2} y^{2}} = 0
\] |
[_exact] |
✓ |
0.378 |
|
|
\[
{}{\mathrm e}^{x} \left (x +1\right )+\left (y \,{\mathrm e}^{y}-x \,{\mathrm e}^{x}\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
0.300 |
|
|
\[
{}\frac {y x +1}{y}+\frac {\left (2 y-x \right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
0.329 |
|
|
\[
{}y^{2}-3 y x -2 x^{2}+\left (y x -x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.466 |
|
|
\[
{}y \left (y+2 x +1\right )-x \left (x +2 y-1\right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.346 |
|
|
\[
{}y \left (2 x -y-1\right )+x \left (2 y-x -1\right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.348 |
|
|
\[
{}y^{2}+12 x^{2} y+\left (2 y x +4 x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.378 |
|
|
\[
{}3 \left (x +y\right )^{2}+x \left (2 x +3 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.491 |
|
|
\[
{}y-\left (y^{2}+x^{2}+x \right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.086 |
|
|
\[
{}2 y x +\left (a +x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
0.180 |
|
|
\[
{}2 y x +x^{2}+b +\left (a +x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
0.207 |
|
|
\[
{}y^{\prime } x +y = x^{3}
\] |
[_linear] |
✓ |
1.128 |
|
|
\[
{}y^{\prime }+a y = b
\] |
[_quadrature] |
✓ |
0.349 |
|