|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = \left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.602 |
|
|
\[
{}\left (-y+x y^{\prime }\right ) \left (x +y y^{\prime }\right ) = h^{2} y^{\prime }
\] |
[_rational] |
✓ |
122.871 |
|
|
\[
{}y^{2} x^{2}-3 x y y^{\prime } = 2 y^{2}+x^{3}
\] |
[_rational, _Bernoulli] |
✓ |
11.544 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }+a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.370 |
|
|
\[
{}y^{2}-2 x y y^{\prime }+{y^{\prime }}^{2} \left (x^{2}-1\right ) = m
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.532 |
|
|
\[
{}y = x y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.393 |
|
|
\[
{}4 {y^{\prime }}^{2} = 9 x
\] |
[_quadrature] |
✓ |
0.318 |
|
|
\[
{}4 x \left (-1+x \right ) \left (-2+x \right ) {y^{\prime }}^{2}-\left (3 x^{2}-6 x +2\right )^{2} = 0
\] |
[_quadrature] |
✓ |
0.308 |
|
|
\[
{}\left (8 {y^{\prime }}^{3}-27\right ) x = \frac {12 {y^{\prime }}^{2}}{x}
\] |
[_quadrature] |
✓ |
0.776 |
|
|
\[
{}3 y = 2 x y^{\prime }-\frac {2 {y^{\prime }}^{2}}{x}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.281 |
|
|
\[
{}y^{2}+{y^{\prime }}^{2} = 1
\] |
[_quadrature] |
✓ |
0.701 |
|
|
\[
{}{y^{\prime }}^{2} \left (2-3 y\right )^{2} = 4-4 y
\] |
[_quadrature] |
✓ |
0.386 |
|
|
\[
{}4 x {y^{\prime }}^{2} = \left (3 x -1\right )^{2}
\] |
[_quadrature] |
✓ |
0.348 |
|
|
\[
{}x {y^{\prime }}^{2}-\left (x -a \right )^{2} = 0
\] |
[_quadrature] |
✓ |
0.369 |
|
|
\[
{}y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.645 |
|
|
\[
{}3 x {y^{\prime }}^{2}-6 y y^{\prime }+x +2 y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.864 |
|
|
\[
{}{y^{\prime }}^{2}+2 x^{3} y^{\prime }-4 x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.648 |
|
|
\[
{}y^{2} \left (y-x y^{\prime }\right ) = x^{4} {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.839 |
|
|
\[
{}{y^{\prime }}^{2} \left (-a^{2}+x^{2}\right )-2 x y y^{\prime }-x^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
31.768 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (x y^{\prime }-2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.705 |
|
|
\[
{}\left (1-y^{2}\right ) {y^{\prime }}^{2} = 1
\] |
[_quadrature] |
✓ |
30.785 |
|
|
\[
{}x^{2}+y = {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
1.434 |
|
|
\[
{}{y^{\prime }}^{3} = y^{4} \left (x y^{\prime }+y\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
13.510 |
|
|
\[
{}\left (1-y^{\prime }\right )^{2}-{\mathrm e}^{-2 y} = {y^{\prime }}^{2} {\mathrm e}^{-2 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
17.747 |
|
|
\[
{}a x y {y^{\prime }}^{2}+\left (x^{2}-a y^{2}-b \right ) y^{\prime }-x y = 0
\] |
[_rational] |
✓ |
116.487 |
|
|
\[
{}{y^{\prime }}^{2} = \left (4 y+1\right ) \left (y^{\prime }-y\right )
\] |
[_quadrature] |
✓ |
1.024 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+b^{2}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.984 |
|
|
\[
{}x y {y^{\prime }}^{2}-\left (x^{2}+y^{2}-1\right ) y^{\prime }+x y = 0
\] |
[_rational] |
✓ |
53.674 |
|
|
\[
{}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}-h^{2}\right ) y^{\prime }-x y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
20.364 |
|
|
\[
{}8 {y^{\prime }}^{3} x = y \left (12 {y^{\prime }}^{2}-9\right )
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.778 |
|
|
\[
{}4 {y^{\prime }}^{2} x^{2} \left (-1+x \right )-4 y^{\prime } x y \left (4 x -3\right )+\left (16 x -9\right ) y^{2} = 0
\] |
[_separable] |
✓ |
0.596 |
|
|
\[
{}\left (y^{2}+x^{2} y^{\prime }\right ) \left (x y^{\prime }+y\right ) = \left (1+y^{\prime }\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
54.421 |
|
|
\[
{}y-x y^{\prime } = a \left (y^{2}+y^{\prime }\right )
\] |
[_separable] |
✓ |
1.520 |
|
|
\[
{}y-x y^{\prime } = b \left (1+x^{2} y^{\prime }\right )
\] |
[_separable] |
✓ |
1.022 |
|
|
\[
{}\left (-y+x y^{\prime }\right ) \left (x -y y^{\prime }\right ) = 2 y^{\prime }
\] |
[_rational] |
✓ |
151.739 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.863 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+y = 2 \ln \left (x \right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.657 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }-2 y = 0
\] |
[[_3rd_order, _exact, _linear, _homogeneous]] |
✓ |
0.107 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }-2 y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.166 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = \ln \left (x \right )^{2}
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.315 |
|
|
\[
{}y^{\prime \prime \prime }-\frac {4 y^{\prime \prime }}{x}+\frac {5 y^{\prime }}{x^{2}}-\frac {2 y}{x^{3}} = 1
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.444 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+x y^{\prime \prime }-4 y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.168 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+7 x y^{\prime }-8 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.110 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+5 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
4.181 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+3 x y^{\prime \prime }+2 y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.335 |
|
|
\[
{}x^{2} y^{\prime \prime }+y = 3 x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.217 |
|
|
\[
{}x^{2} y^{\prime \prime }+7 x y^{\prime }+5 y = x^{5}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.580 |
|
|
\[
{}x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y = x^{4}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.369 |
|
|
\[
{}x^{2} y^{\prime \prime }-4 x y^{\prime }+6 y = x^{4}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.367 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }-4 y = x^{4}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.424 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{m}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.180 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y = x^{m}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.505 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime } = \ln \left (x \right )
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.940 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y = {\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.923 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.400 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+3 x y^{\prime \prime }+2 y^{\prime } = x
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.438 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = 4 x
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
0.751 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+8 x y^{\prime }+2 y = x^{2}+3 x -4
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.579 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }-20 y = \left (x +1\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.168 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+7 x y^{\prime }-8 y = x^{2}+\frac {1}{x^{2}}
\] |
[[_3rd_order, _reducible, _mu_y2]] |
✓ |
0.348 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+2 x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-x y^{\prime }+y = x +\ln \left (x \right )
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
0.401 |
|
|
\[
{}x^{2} y^{\prime \prime }-x y^{\prime }+2 y = \ln \left (x \right ) x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
126.086 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 x y^{\prime }+5 y = x^{2} \sin \left (\ln \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
264.578 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-x y^{\prime }+y = \ln \left (x \right ) x
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.297 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+3 x y^{\prime }+y = \left (1+\ln \left (x \right )\right )^{2}
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.828 |
|
|
\[
{}\left (5+2 x \right )^{2} y^{\prime \prime }-6 \left (5+2 x \right ) y^{\prime }+8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.316 |
|
|
\[
{}\left (x +1\right )^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime } = \left (2 x +3\right ) \left (2 x +4\right )
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.998 |
|
|
\[
{}x y^{\prime \prime }+2 x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.972 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{x} \left (y^{\prime }+y\right ) = {\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.987 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.325 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+8 x y^{\prime }+2 y = x^{2}+3 x -4
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.490 |
|
|
\[
{}x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 x y^{\prime }+2 y = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.348 |
|
|
\[
{}y^{\prime \prime }+2 \,{\mathrm e}^{x} y^{\prime }+2 y \,{\mathrm e}^{x} = x^{2}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.705 |
|
|
\[
{}\left (x^{2}-x \right ) y^{\prime \prime }+2 \left (2 x +1\right ) y^{\prime }+2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.309 |
|
|
\[
{}\left (x^{2}-x \right ) y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }-4 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.058 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y = 2 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
3.039 |
|
|
\[
{}\left (2 x^{2}+3 x \right ) y^{\prime \prime }+\left (3+6 x \right ) y^{\prime }+2 y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.295 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}+y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.893 |
|
|
\[
{}\left (-b \,x^{2}+a x \right ) y^{\prime \prime }+2 a y^{\prime }+2 b y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.277 |
|
|
\[
{}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.509 |
|
|
\[
{}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y = 2
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
0.040 |
|
|
\[
{}x^{5} y^{\left (6\right )}+x^{4} y^{\left (5\right )}+x y^{\prime }+y = \ln \left (x \right )
\] |
[[_high_order, _exact, _linear, _nonhomogeneous]] |
✗ |
0.439 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }+4 x^{2} y^{\prime \prime }+x \left (x^{2}+2\right ) y^{\prime }+3 x^{2} y = 2 x
\] |
[[_3rd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.428 |
|
|
\[
{}x^{5} y^{\prime \prime }+3 x^{3} y^{\prime }+\left (3-6 x \right ) x^{2} y = x^{4}+2 x -5
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.593 |
|
|
\[
{}y^{\prime \prime \prime } = f \left (x \right )
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.154 |
|
|
\[
{}y^{2}+\left (2 x y-1\right ) y^{\prime }+x y^{\prime \prime }+x^{2} y^{\prime \prime \prime } = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✗ |
0.035 |
|
|
\[
{}y^{\prime \prime } = x +\sin \left (x \right )
\] |
[[_2nd_order, _quadrature]] |
✓ |
2.049 |
|
|
\[
{}y^{\prime \prime } = x \,{\mathrm e}^{x}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.835 |
|
|
\[
{}y^{\prime \prime } \cos \left (x \right )^{2} = 1
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.053 |
|
|
\[
{}x^{3} y^{\prime \prime \prime } = 1
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.174 |
|
|
\[
{}y^{\prime \prime } = \frac {a}{x}
\] |
[[_2nd_order, _quadrature]] |
✓ |
1.756 |
|
|
\[
{}y^{\prime \prime \prime } \csc \left (x \right )^{2} = 1
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.235 |
|
|
\[
{}y^{\prime \prime } \sqrt {a^{2}+x^{2}} = x
\] |
[[_2nd_order, _quadrature]] |
✓ |
0.652 |
|
|
\[
{}x^{2} y^{\prime \prime } = \ln \left (x \right )
\] |
[[_2nd_order, _quadrature]] |
✓ |
0.494 |
|
|
\[
{}y^{\prime \prime } = y
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.917 |
|
|
\[
{}y^{3} y^{\prime \prime } = a
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
2.050 |
|
|
\[
{}y^{\prime \prime }-a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.536 |
|
|
\[
{}y^{\prime \prime }+\frac {a^{2}}{y} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.865 |
|
|
\[
{}y^{\prime \prime } = y^{3}-y
\] |
[[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
4.412 |
|
|
\[
{}y^{\prime \prime } = {\mathrm e}^{2 y}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.597 |
|