|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y = x \left (1+y^{\prime }\right )+{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.408 |
|
|
\[
{}y = 2 y^{\prime } x +\sin \left (y^{\prime }\right )
\] |
[_dAlembert] |
✓ |
0.988 |
|
|
\[
{}y = {y^{\prime }}^{2} x -\frac {1}{y^{\prime }}
\] |
[_dAlembert] |
✓ |
3.174 |
|
|
\[
{}y = \frac {3 y^{\prime } x}{2}+{\mathrm e}^{y^{\prime }}
\] |
[_dAlembert] |
✓ |
0.994 |
|
|
\[
{}y = y^{\prime } x +\frac {a}{{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.743 |
|
|
\[
{}y = y^{\prime } x +{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.320 |
|
|
\[
{}{y^{\prime }}^{2} x -y y^{\prime }-y^{\prime }+1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.397 |
|
|
\[
{}y = y^{\prime } x +a \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
6.171 |
|
|
\[
{}x = \frac {y}{y^{\prime }}+\frac {1}{{y^{\prime }}^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
0.334 |
|
|
\[
{}y^{\prime } {\mathrm e}^{-x}+y^{2}-2 y \,{\mathrm e}^{x} = 1-{\mathrm e}^{2 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.197 |
|
|
\[
{}y^{\prime }+y^{2}-2 y \sin \left (x \right )+\sin \left (x \right )^{2}-\cos \left (x \right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.529 |
|
|
\[
{}y^{\prime } x -y^{2}+\left (2 x +1\right ) y = x^{2}+2 x
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
1.527 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+y x +1
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.269 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) y^{2}-4 y y^{\prime }-4 x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
4.889 |
|
|
\[
{}{y^{\prime }}^{2}-4 y = 0
\] |
[_quadrature] |
✓ |
0.520 |
|
|
\[
{}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
8.317 |
|
|
\[
{}{y^{\prime }}^{2}-y^{2} = 0
\] |
[_quadrature] |
✓ |
0.829 |
|
|
\[
{}y^{\prime } = y^{{2}/{3}}+a
\] |
[_quadrature] |
✓ |
0.645 |
|
|
\[
{}\left (y^{\prime } x +y\right )^{2}+3 x^{5} \left (y^{\prime } x -2 y\right ) = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
13.506 |
|
|
\[
{}y \left (y-2 y^{\prime } x \right )^{2} = 2 y^{\prime }
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.576 |
|
|
\[
{}8 {y^{\prime }}^{3}-12 {y^{\prime }}^{2} = -27 x +27 y
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.685 |
|
|
\[
{}\left (y^{\prime }-1\right )^{2} = y^{2}
\] |
[_quadrature] |
✓ |
0.658 |
|
|
\[
{}y = {y^{\prime }}^{2}-y^{\prime } x +x
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.449 |
|
|
\[
{}\left (y^{\prime } x +y\right )^{2} = y^{2} y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
150.990 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+y^{2} = 1
\] |
[_quadrature] |
✓ |
1.280 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.594 |
|
|
\[
{}3 {y^{\prime }}^{2} x -6 y y^{\prime }+x +2 y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
2.291 |
|
|
\[
{}y = y^{\prime } x +\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
7.415 |
|
|
\[
{}y^{\prime } = \left (x -y\right )^{2}+1
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
2.267 |
|
|
\[
{}x \sin \left (x \right ) y^{\prime }+\left (\sin \left (x \right )-x \cos \left (x \right )\right ) y = \sin \left (x \right ) \cos \left (x \right )-x
\] |
[_linear] |
✓ |
9.731 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = y^{n} \sin \left (2 x \right )
\] |
[_Bernoulli] |
✓ |
4.345 |
|
|
\[
{}x^{3}-3 x y^{2}+\left (y^{3}-3 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
14.684 |
|
|
\[
{}5 y x -4 y^{2}-6 x^{2}+\left (y^{2}-8 y x +\frac {5 x^{2}}{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
129.225 |
|
|
\[
{}3 x y^{2}-x^{2}+\left (3 x^{2} y-6 y^{2}-1\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.477 |
|
|
\[
{}y-x y^{2} \ln \left (x \right )+y^{\prime } x = 0
\] |
[_Bernoulli] |
✓ |
2.353 |
|
|
\[
{}2 x y \,{\mathrm e}^{x^{2}}-x \sin \left (x \right )+{\mathrm e}^{x^{2}} y^{\prime } = 0
\] |
[_linear] |
✓ |
2.449 |
|
|
\[
{}y^{\prime } = \frac {1}{2 x -y^{2}}
\] |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
0.942 |
|
|
\[
{}x^{2}+y^{\prime } x = 3 x +y^{\prime }
\] |
[_quadrature] |
✓ |
0.319 |
|
|
\[
{}x y y^{\prime }-y^{2} = x^{4}
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
1.939 |
|
|
\[
{}\frac {1}{y^{2}-y x +x^{2}} = \frac {y^{\prime }}{2 y^{2}-y x}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
17.528 |
|
|
\[
{}\left (2 x -1\right ) y^{\prime }-2 y = \frac {1-4 x}{x^{2}}
\] |
[_linear] |
✓ |
1.027 |
|
|
\[
{}x -y+3+\left (3 x +y+1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.802 |
|
|
\[
{}y^{\prime }+\cos \left (\frac {x}{2}+\frac {y}{2}\right ) = \cos \left (\frac {x}{2}-\frac {y}{2}\right )
\] |
[_separable] |
✓ |
5.882 |
|
|
\[
{}y^{\prime } \left (3 x^{2}-2 x \right )-y \left (6 x -2\right ) = 0
\] |
[_separable] |
✓ |
1.257 |
|
|
\[
{}x y^{2} y^{\prime }-y^{3} = \frac {x^{4}}{3}
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.474 |
|
|
\[
{}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
4.596 |
|
|
\[
{}x^{2}+y^{2}-x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
4.239 |
|
|
\[
{}x -y+2+\left (x -y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.176 |
|
|
\[
{}x y^{2}+y-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
1.813 |
|
|
\[
{}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
1.837 |
|
|
\[
{}\left (x -1\right ) \left (y^{2}-y+1\right ) = \left (-1+y\right ) \left (x^{2}+x +1\right ) y^{\prime }
\] |
[_separable] |
✓ |
1.915 |
|
|
\[
{}\left (x -2 y x -y^{2}\right ) y^{\prime }+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.523 |
|
|
\[
{}y \cos \left (x \right )+\left (2 y-\sin \left (x \right )\right ) y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.295 |
|
|
\[
{}y^{\prime }-1 = {\mathrm e}^{x +2 y}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.605 |
|
|
\[
{}2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.140 |
|
|
\[
{}x^{2} y^{n} y^{\prime } = 2 y^{\prime } x -y
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.007 |
|
|
\[
{}\left (3 x +3 y+a^{2}\right ) y^{\prime } = 4 x +4 y+b^{2}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.245 |
|
|
\[
{}x -y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.491 |
|
|
\[
{}y^{\prime } x +y = y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.333 |
|
|
\[
{}\sin \left (\ln \left (x \right )\right )-\cos \left (\ln \left (y\right )\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.152 |
|
|
\[
{}y^{\prime } = \sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.268 |
|
|
\[
{}\left (5 x -7 y+1\right ) y^{\prime }+x +y-1 = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.465 |
|
|
\[
{}x +y+1+\left (2 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.530 |
|
|
\[
{}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.692 |
|
|
\[
{}y^{\prime } = \frac {2 \left (2+y\right )^{2}}{\left (x +y-1\right )^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
1.672 |
|
|
\[
{}4 x^{2} {y^{\prime }}^{2}-y^{2} = x y^{3}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.627 |
|
|
\[
{}y^{\prime }+{y^{\prime }}^{2} x -y = 0
\] |
[_rational, _dAlembert] |
✓ |
0.853 |
|
|
\[
{}y^{\prime \prime }+y = 2 \cos \left (x \right )+2 \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.480 |
|
|
\[
{}x y^{\prime \prime \prime } = 2
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.186 |
|
|
\[
{}y^{\prime \prime } = {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.178 |
|
|
\[
{}\left (x -1\right ) y^{\prime \prime } = 1
\] |
[[_2nd_order, _quadrature]] |
✓ |
0.911 |
|
|
\[
{}{y^{\prime }}^{4} = 1
\] |
[_quadrature] |
✓ |
0.863 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.693 |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }+2 y = 2
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.860 |
|
|
\[
{}y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.461 |
|
|
\[
{}{y^{\prime }}^{2}+y y^{\prime \prime } = 1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.185 |
|
|
\[
{}y^{\prime \prime \prime \prime } = x
\] |
[[_high_order, _quadrature]] |
✓ |
0.105 |
|
|
\[
{}y^{\prime \prime \prime } = x +\cos \left (x \right )
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.139 |
|
|
\[
{}y^{\prime \prime } \left (x +2\right )^{5} = 1
\] |
[[_2nd_order, _quadrature]] |
✓ |
0.921 |
|
|
\[
{}y^{\prime \prime } = x \,{\mathrm e}^{x}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.505 |
|
|
\[
{}y^{\prime \prime } = 2 x \ln \left (x \right )
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.263 |
|
|
\[
{}x y^{\prime \prime } = y^{\prime }
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.905 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.840 |
|
|
\[
{}x y^{\prime \prime } = \left (2 x^{2}+1\right ) y^{\prime }
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.892 |
|
|
\[
{}x y^{\prime \prime } = y^{\prime }+x^{2}
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.105 |
|
|
\[
{}x \ln \left (x \right ) y^{\prime \prime } = y^{\prime }
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.728 |
|
|
\[
{}y x = y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right )
\] |
[_separable] |
✓ |
2.340 |
|
|
\[
{}2 y^{\prime \prime } = \frac {y^{\prime }}{x}+\frac {x^{2}}{y^{\prime }}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
0.672 |
|
|
\[
{}y^{\prime \prime \prime } = \sqrt {1-{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
0.356 |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.166 |
|
|
\[
{}y^{\prime \prime } = \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.369 |
|
|
\[
{}y^{\prime \prime } = {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.177 |
|
|
\[
{}y^{\prime \prime } = \sqrt {1-{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.369 |
|
|
\[
{}y^{\prime \prime } = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.355 |
|
|
\[
{}y^{\prime \prime } = \sqrt {1+y^{\prime }}
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.339 |
|
|
\[
{}y^{\prime \prime } = y^{\prime } \ln \left (y^{\prime }\right )
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.218 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+2 = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.671 |
|
|
\[
{}y^{\prime \prime } = y^{\prime } \left (1+y^{\prime }\right )
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.434 |
|
|
\[
{}3 y^{\prime \prime } = \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.819 |
|
|
\[
{}y^{\prime \prime \prime }+{y^{\prime \prime }}^{2} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✓ |
0.135 |
|