|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\frac {x +y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y-x y^{\prime }}{y^{2}+x^{2}} = 0
\] |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
2.855 |
|
|
\[
{}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
4.142 |
|
|
\[
{}\frac {\sin \left (\frac {x}{y}\right )}{y}-\frac {y \cos \left (\frac {y}{x}\right )}{x^{2}}+1+\left (\frac {\cos \left (\frac {y}{x}\right )}{x}-\frac {x \sin \left (\frac {x}{y}\right )}{y^{2}}+\frac {1}{y^{2}}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
37.873 |
|
|
\[
{}\frac {1}{x}-\frac {y^{2}}{\left (x -y\right )^{2}}+\left (\frac {x^{2}}{\left (x -y\right )^{2}}-\frac {1}{y}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.845 |
|
|
\[
{}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.785 |
|
|
\[
{}\left (y^{2} x^{2}-1\right ) y^{\prime }+2 x y^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.394 |
|
|
\[
{}a x y^{\prime }+b y+x^{m} y^{n} \left (\alpha x y^{\prime }+\beta y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.349 |
|
|
\[
{}2 x y^{2}-y+\left (y^{2}+x +y\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.392 |
|
|
\[
{}y^{\prime } = 2 x y-x^{3}+x
\] |
[_linear] |
✓ |
1.375 |
|
|
\[
{}x y^{\prime }+y-x y^{2} \ln \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.120 |
|
|
\[
{}2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
2.673 |
|
|
\[
{}{y^{\prime }}^{2} y+y^{\prime } \left (x -y\right )-x = 0
\] |
[_quadrature] |
✓ |
3.211 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2} = x^{4}+y^{2} x^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
9.345 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x^{3} y+y^{2} x^{2}+x y^{3}\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
2.133 |
|
|
\[
{}x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.174 |
|
|
\[
{}x {y^{\prime }}^{3} = 1+y^{\prime }
\] |
[_quadrature] |
✓ |
0.514 |
|
|
\[
{}{y^{\prime }}^{3}-x^{3} \left (1-y^{\prime }\right ) = 0
\] |
[_quadrature] |
✓ |
0.503 |
|
|
\[
{}{y^{\prime }}^{3}+y^{3}-3 y y^{\prime } = 0
\] |
[_quadrature] |
✓ |
19.495 |
|
|
\[
{}y = {\mathrm e}^{y^{\prime }} {y^{\prime }}^{2}
\] |
[_quadrature] |
✓ |
1.638 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
3.259 |
|
|
\[
{}y \left (1+{y^{\prime }}^{2}\right ) = 2 \alpha
\] |
[_quadrature] |
✓ |
0.497 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (x y^{\prime }-2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.735 |
|
|
\[
{}y = 2 x y^{\prime }+\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
10.149 |
|
|
\[
{}y = \frac {k \left (x +y y^{\prime }\right )}{\sqrt {1+{y^{\prime }}^{2}}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
12.771 |
|
|
\[
{}x = y y^{\prime }+a {y^{\prime }}^{2}
\] |
unknown |
✓ |
78.141 |
|
|
\[
{}y = x {y^{\prime }}^{2}+{y^{\prime }}^{3}
\] |
unknown |
✓ |
2.853 |
|
|
\[
{}y = x y^{\prime }+y^{\prime }-{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.382 |
|
|
\[
{}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
106.559 |
|
|
\[
{}{y^{\prime }}^{2} \left (x^{2}-1\right )-2 x y y^{\prime }+y^{2}-1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
0.583 |
|
|
\[
{}{y^{\prime }}^{2}+2 x y^{\prime }+2 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.464 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
2.040 |
|
|
\[
{}y^{\prime } = \sqrt {y-x}+1
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.274 |
|
|
\[
{}y^{\prime } = \sqrt {y}
\] |
[_quadrature] |
✓ |
1.314 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )
\] |
[_quadrature] |
✓ |
2.397 |
|
|
\[
{}y^{\prime } = y \ln \left (y\right )^{2}
\] |
[_quadrature] |
✓ |
7.073 |
|
|
\[
{}y^{\prime } = -x +\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.060 |
|
|
\[
{}y^{\prime } = -x -\sqrt {x^{2}+2 y}
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
2.019 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }+4 x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.596 |
|
|
\[
{}x {y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.319 |
|
|
\[
{}y^{2} \left (y^{\prime }-1\right ) = \left (2-y^{\prime }\right )^{2}
\] |
[_quadrature] |
✓ |
3.365 |
|
|
\[
{}{y^{\prime }}^{4} = 4 y \left (x y^{\prime }-2 y\right )^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.747 |
|
|
\[
{}x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 x y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.685 |
|
|
\[
{}y = {y^{\prime }}^{2}-x y^{\prime }+\frac {x^{3}}{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
4.791 |
|
|
\[
{}y = 2 x y^{\prime }+\frac {x^{2}}{2}+{y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
10.826 |
|
|
\[
{}{y^{\prime }}^{2}-y y^{\prime }+{\mathrm e}^{x} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.563 |
|
|
\[
{}{y^{\prime \prime \prime }}^{2}+x^{2} = 1
\] |
[[_3rd_order, _quadrature]] |
✓ |
0.713 |
|
|
\[
{}y^{\prime \prime } = \frac {1}{\sqrt {y}}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.611 |
|
|
\[
{}a^{3} y^{\prime \prime \prime } y^{\prime \prime } = \sqrt {1+c^{2} {y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
19.360 |
|
|
\[
{}y^{\prime \prime \prime } = \sqrt {1+{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
0.386 |
|
|
\[
{}2 \left (2 a -y\right ) y^{\prime \prime } = 1+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.185 |
|
|
\[
{}y^{\prime \prime }-x y^{\prime \prime \prime }+{y^{\prime \prime \prime }}^{3} = 0
\] |
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]] |
✓ |
0.545 |
|
|
\[
{}y y^{\prime \prime }+{y^{\prime }}^{2} = y^{2} \ln \left (y\right )
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.695 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.250 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.800 |
|
|
\[
{}n \,x^{3} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.168 |
|
|
\[
{}y^{2} \left (x^{2} y^{\prime \prime }-x y^{\prime }+y\right ) = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.159 |
|
|
\[
{}x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.152 |
|
|
\[
{}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0
\] |
[[_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]] |
✗ |
3.167 |
|
|
\[
{}x \left (x^{2} y^{\prime }+2 x y\right ) y^{\prime \prime }+4 x {y^{\prime }}^{2}+8 x y y^{\prime }+4 y^{2}-1 = 0
\] |
[NONE] |
✗ |
0.146 |
|
|
\[
{}x \left (x y+1\right ) y^{\prime \prime }+x^{2} {y^{\prime }}^{2}+\left (4 x y+2\right ) y^{\prime }+y^{2}+1 = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.836 |
|
|
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.814 |
|
|
\[
{}a^{2} y^{\prime \prime } = 2 x \sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
1.154 |
|
|
\[
{}x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime } = 4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.196 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.487 |
|
|
\[
{}5 {y^{\prime \prime \prime }}^{2}-3 y^{\prime \prime } y^{\prime \prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries], [_high_order, _reducible, _mu_poly_yn]] |
✓ |
0.676 |
|
|
\[
{}40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )} = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✗ |
0.087 |
|
|
\[
{}{y^{\prime \prime }}^{2}+2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.656 |
|
|
\[
{}{y^{\prime \prime }}^{2}-2 x y^{\prime \prime }-y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.937 |
|
|
\[
{}2 x^{3} y^{\prime \prime \prime }-6 x^{2} y^{\prime \prime }+12 x y^{\prime }-12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.127 |
|
|
\[
{}y^{\prime \prime \prime }-\frac {3 y^{\prime \prime }}{x}+\frac {6 y^{\prime }}{x^{2}}-\frac {6 y}{x^{3}} = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✓ |
0.128 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✗ |
0.943 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}+y = 0
\] |
[_Lienard] |
✓ |
1.632 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2} = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.706 |
|
|
\[
{}x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }+6 x y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.129 |
|
|
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.050 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.277 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 2 x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.769 |
|
|
\[
{}y^{\prime \prime }+\frac {x y^{\prime }}{1-x}-\frac {y}{1-x} = x -1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.850 |
|
|
\[
{}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = x^{4}+12
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
0.056 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime } = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.066 |
|
|
\[
{}y^{\prime \prime }+y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.853 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{x^{2} \ln \left (x \right )} = {\mathrm e}^{x} \left (\frac {2}{x}+\ln \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
0.265 |
|
|
\[
{}y^{\prime \prime }+p_{1} y^{\prime }+p_{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.124 |
|
|
\[
{}\left (2 x +1\right ) y^{\prime \prime }+\left (4 x -2\right ) y^{\prime }-8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.165 |
|
|
\[
{}y^{\prime \prime } \sin \left (x \right )^{2}+\sin \left (x \right ) \cos \left (x \right ) y^{\prime } = y
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.017 |
|
|
\[
{}y^{\prime \prime \prime \prime }-2 y^{\prime \prime } = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.070 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.068 |
|
|
\[
{}y^{\prime \prime \prime \prime }+4 y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.081 |
|
|
\[
{}y^{\prime \prime \prime \prime }-y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.074 |
|
|
\[
{}2 y^{\prime \prime }+y^{\prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.856 |
|
|
\[
{}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }+y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.097 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.097 |
|
|
\[
{}y^{\prime \prime }-6 y^{\prime }+8 y = {\mathrm e}^{x}+{\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.172 |
|
|
\[
{}y^{\prime \prime \prime }+y^{\prime \prime }+y^{\prime }+y = x \,{\mathrm e}^{x}
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.135 |
|
|
\[
{}y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.156 |
|
|
\[
{}y^{\prime \prime }+4 y = x \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
4.852 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }+y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
32.729 |
|
|
\[
{}y^{\prime \prime }-y = \frac {{\mathrm e}^{x}-{\mathrm e}^{-x}}{{\mathrm e}^{x}+{\mathrm e}^{-x}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.742 |
|
|
\[
{}y^{\prime \prime }-2 y = 4 x^{2} {\mathrm e}^{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.303 |
|
|
\[
{}y^{\prime \prime }+y = \sin \left (x \right ) \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.860 |
|