|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.984 |
|
|
\[
{}3 y-7 x +7 = \left (3 x -7 y-3\right ) y^{\prime }
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
4.018 |
|
|
\[
{}\left (x +2 y+1\right ) y^{\prime } = 2 x +4 y+3
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.849 |
|
|
\[
{}y^{\prime } = \frac {2 \left (y+2\right )^{2}}{\left (x +y-1\right )^{2}}
\] |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
1.829 |
|
|
\[
{}\left (x +y\right )^{2} y^{\prime } = a^{2}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
3.278 |
|
|
\[
{}y^{\prime } x -4 y = x^{2} \sqrt {y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
3.395 |
|
|
\[
{}\cos \left (x \right ) y^{\prime } = y \sin \left (x \right )+\cos \left (x \right )^{2}
\] |
[_linear] |
✓ |
2.257 |
|
|
\[
{}y^{\prime } = 2 x y-x^{3}+x
\] |
[_linear] |
✓ |
1.606 |
|
|
\[
{}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{x \left (x^{2}+1\right )}
\] |
[_linear] |
✓ |
1.264 |
|
|
\[
{}\left (x -2 x y-y^{2}\right ) y^{\prime }+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.858 |
|
|
\[
{}y^{\prime } x +y = x y^{2} \ln \left (x \right )
\] |
[_Bernoulli] |
✓ |
2.186 |
|
|
\[
{}y^{\prime }-\frac {x y}{2 x^{2}-2}-\frac {x}{2 y} = 0
\] |
[_rational, _Bernoulli] |
✓ |
1.779 |
|
|
\[
{}y^{\prime } \left (x^{2} y^{3}+x y\right ) = 1
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.466 |
|
|
\[
{}x -y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.841 |
|
|
\[
{}y^{\prime } = \frac {y^{2}}{3}+\frac {2}{3 x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
1.836 |
|
|
\[
{}y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.115 |
|
|
\[
{}y^{\prime } x -3 y+y^{2} = 4 x^{2}-4 x
\] |
[_rational, _Riccati] |
✓ |
1.532 |
|
|
\[
{}y^{\prime } = y^{2}+\frac {1}{x^{4}}
\] |
[_rational, [_Riccati, _special]] |
✓ |
1.460 |
|
|
\[
{}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime } = \left (1+y^{2}\right )^{{3}/{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
5.148 |
|
|
\[
{}y^{\prime } \left (x^{2}+y^{2}+3\right ) = 2 x \left (2 y-\frac {x^{2}}{y}\right )
\] |
[_rational] |
✗ |
2.846 |
|
|
\[
{}y^{\prime } = \frac {x -y^{2}}{2 y \left (x +y^{2}\right )}
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.723 |
|
|
\[
{}\left (\left (x +y\right ) x +a^{2}\right ) y^{\prime } = y \left (x +y\right )+b^{2}
\] |
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.323 |
|
|
\[
{}y^{\prime } = k y+f \left (x \right )
\] |
[[_linear, ‘class A‘]] |
✓ |
1.185 |
|
|
\[
{}y^{\prime } = y^{2}-x^{2}
\] |
[_Riccati] |
✓ |
1.047 |
|
|
\[
{}\frac {x +y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y-y^{\prime } x}{x^{2}+y^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
2.751 |
|
|
\[
{}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.339 |
|
|
\[
{}\frac {\sin \left (\frac {x}{y}\right )}{y}-\frac {y \cos \left (\frac {y}{x}\right )}{x^{2}}+1+\left (\frac {\cos \left (\frac {y}{x}\right )}{x}-\frac {x \sin \left (\frac {x}{y}\right )}{y^{2}}+\frac {1}{y^{2}}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
38.013 |
|
|
\[
{}\frac {1}{x}-\frac {y^{2}}{\left (x -y\right )^{2}}+\left (\frac {x^{2}}{\left (x -y\right )^{2}}-\frac {1}{y}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.911 |
|
|
\[
{}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.941 |
|
|
\[
{}\left (x^{2} y^{2}-1\right ) y^{\prime }+2 x y^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.526 |
|
|
\[
{}a x y^{\prime }+b y+x^{m} y^{n} \left (\alpha x y^{\prime }+\beta y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.133 |
|
|
\[
{}2 x y^{2}-y+\left (y^{2}+x +y\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.396 |
|
|
\[
{}y^{\prime } = 2 x y-x^{3}+x
\] |
[_linear] |
✓ |
1.601 |
|
|
\[
{}y^{\prime } x +y-x y^{2} \ln \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.236 |
|
|
\[
{}2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
2.632 |
|
|
\[
{}{y^{\prime }}^{2} y+y^{\prime } \left (x -y\right )-x = 0
\] |
[_quadrature] |
✓ |
4.015 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = x^{4}+x^{2} y^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
9.350 |
|
|
\[
{}{y^{\prime }}^{3}-\left (x^{2}+x y+y^{2}\right ) {y^{\prime }}^{2}+\left (x^{3} y+x^{2} y^{2}+x y^{3}\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
3.115 |
|
|
\[
{}x {y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.114 |
|
|
\[
{}x {y^{\prime }}^{3} = 1+y^{\prime }
\] |
[_quadrature] |
✓ |
0.503 |
|
|
\[
{}{y^{\prime }}^{3}-x^{3} \left (1-y^{\prime }\right ) = 0
\] |
[_quadrature] |
✓ |
0.450 |
|
|
\[
{}{y^{\prime }}^{3}+y^{3}-3 y y^{\prime } = 0
\] |
[_quadrature] |
✓ |
18.616 |
|
|
\[
{}y = {\mathrm e}^{y^{\prime }} {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
1.560 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
3.192 |
|
|
\[
{}y \left ({y^{\prime }}^{2}+1\right ) = 2 \alpha
\] |
[_quadrature] |
✓ |
0.483 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (y^{\prime } x -2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.606 |
|
|
\[
{}y = 2 y^{\prime } x +\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
9.784 |
|
|
\[
{}y = \frac {k \left (x +y y^{\prime }\right )}{\sqrt {{y^{\prime }}^{2}+1}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
12.332 |
|
|
\[
{}x = y y^{\prime }+a {y^{\prime }}^{2}
\] |
[_dAlembert] |
✓ |
76.267 |
|
|
\[
{}y = x {y^{\prime }}^{2}+{y^{\prime }}^{3}
\] |
[_dAlembert] |
✓ |
2.727 |
|
|
\[
{}y = y^{\prime } x +y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.362 |
|
|
\[
{}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
107.644 |
|
|
\[
{}{y^{\prime }}^{2} \left (x^{2}-1\right )-2 x y y^{\prime }+y^{2}-1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.533 |
|
|
\[
{}{y^{\prime }}^{2}+2 y^{\prime } x +2 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.431 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.790 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}+1
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.144 |
|
|
\[
{}y^{\prime } = \sqrt {y}
\] |
[_quadrature] |
✓ |
1.507 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )
\] |
[_quadrature] |
✓ |
2.441 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )^{2}
\] |
[_quadrature] |
✓ |
6.825 |
|
|
\[
{}y^{\prime } = -x +\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.188 |
|
|
\[
{}y^{\prime } = -x -\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.131 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.442 |
|
|
\[
{}x {y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.106 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
3.193 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (y^{\prime } x -2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.611 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.566 |
|
|
\[
{}y = {y^{\prime }}^{2}-y^{\prime } x +\frac {x^{3}}{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
4.559 |
|
|
\[
{}y = 2 y^{\prime } x +\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
9.694 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.218 |
|
|
\[
{}{y^{\prime \prime \prime }}^{2}+x^{2} = 1
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.513 |
|
|
\[
{}y^{\prime \prime } = \frac {1}{\sqrt {y}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.584 |
|
|
\[
{}a^{3} y^{\prime \prime \prime } y^{\prime \prime } = \sqrt {1+c^{2} {y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
3.287 |
|
|
\[
{}y^{\prime \prime \prime } = \sqrt {1+{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
2.154 |
|
|
\[
{}2 \left (2 a -y\right ) y^{\prime \prime } = {y^{\prime }}^{2}+1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.102 |
|
|
\[
{}y^{\prime \prime }-x y^{\prime \prime \prime }+{y^{\prime \prime \prime }}^{3} = 0
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✓ |
0.541 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2} = y^{2} \ln \left (y\right )
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.575 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.246 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.733 |
|
|
\[
{}n \,x^{3} y^{\prime \prime } = \left (y-y^{\prime } x \right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.164 |
|
|
\[
{}y^{2} \left (x^{2} y^{\prime \prime }-y^{\prime } x +y\right ) = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.158 |
|
|
\[
{}x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.150 |
|
|
\[
{}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
5.878 |
|
|
\[
{}x \left (x^{2} y^{\prime }+2 x y\right ) y^{\prime \prime }+4 x {y^{\prime }}^{2}+8 x y y^{\prime }+4 y^{2}-1 = 0
\] |
[NONE] |
✗ |
0.173 |
|
|
\[
{}x \left (x y+1\right ) y^{\prime \prime }+x^{2} {y^{\prime }}^{2}+\left (4 x y+2\right ) y^{\prime }+y^{2}+1 = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.793 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.757 |
|
|
\[
{}a^{2} y^{\prime \prime } = 2 x \sqrt {{y^{\prime }}^{2}+1}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
1.138 |
|
|
\[
{}x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime } = 4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.149 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.481 |
|
|
\[
{}5 {y^{\prime \prime \prime }}^{2}-3 y^{\prime \prime } y^{\prime \prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] |
✓ |
0.740 |
|
|
\[
{}40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )} = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✗ |
0.091 |
|
|
\[
{}{y^{\prime \prime }}^{2}+2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.639 |
|
|
\[
{}{y^{\prime \prime }}^{2}-2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.593 |
|
|
\[
{}2 x^{3} y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+12 y^{\prime } x -12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.128 |
|
|
\[
{}y^{\prime \prime \prime }-\frac {3 y^{\prime \prime }}{x}+\frac {6 y^{\prime }}{x^{2}}-\frac {6 y}{x^{3}} = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.125 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.930 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+y = 0
\] |
[_Lienard] |
✓ |
1.536 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2} = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.783 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 y^{\prime } x -6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.127 |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime } x -y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.056 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.260 |
|