|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}2 y^{\prime }+x = 4 \sqrt {y}
\] |
[[_1st_order, _with_linear_symmetries], _Chini] |
✓ |
1.098 |
|
|
\[
{}y^{\prime } = y^{2}-\frac {2}{x^{2}}
\] |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
0.543 |
|
|
\[
{}2 y^{\prime } x +y = y^{2} \sqrt {x -x^{2} y^{2}}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
10.108 |
|
|
\[
{}\frac {2 x y y^{\prime }}{3} = \sqrt {x^{6}-y^{4}}+y^{2}
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
4.220 |
|
|
\[
{}2 y+\left (x^{2} y+1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.351 |
|
|
\[
{}y \left (y x +1\right )+\left (1-y x \right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
0.411 |
|
|
\[
{}y \left (x^{2} y^{2}+1\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
0.553 |
|
|
\[
{}\left (x^{2}-y^{4}\right ) y^{\prime }-y x = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.312 |
|
|
\[
{}y \left (1+\sqrt {x^{2} y^{4}-1}\right )+2 y^{\prime } x = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.762 |
|
|
\[
{}x \left (2-9 x y^{2}\right )+y \left (4 y^{2}-6 x^{3}\right ) y^{\prime } = 0
\] |
[_exact, _rational] |
✓ |
1.736 |
|
|
\[
{}\frac {y}{x}+\left (y^{3}+\ln \left (x \right )\right ) y^{\prime } = 0
\] |
[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.481 |
|
|
\[
{}2 x +3+\left (-2+2 y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.309 |
|
|
\[
{}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.013 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }-y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.035 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}} = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.228 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.506 |
|
|
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.831 |
|
|
\[
{}y^{\prime \prime }+\frac {y^{\prime }}{x}+x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.890 |
|
|
\[
{}x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.189 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.858 |
|
|
\[
{}y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.041 |
|
|
\[
{}y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+y \,{\mathrm e}^{x} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.314 |
|
|
\[
{}x^{2} y^{\prime \prime }+2 y^{\prime } x +4 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.122 |
|
|
\[
{}x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
0.301 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.444 |
|
|
\[
{}y^{\prime \prime }+y^{\prime } x +y = 2 x \,{\mathrm e}^{x}-1
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.981 |
|
|
\[
{}x y^{\prime \prime }+y^{\prime } x -y = x^{2}+2 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.468 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x -y = x^{2}+2 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
2.061 |
|
|
\[
{}x^{3} y^{\prime \prime }+y^{\prime } x -y = \cos \left (\frac {1}{x}\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.102 |
|
|
\[
{}x \left (x +1\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.662 |
|
|
\[
{}2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y = x^{2}-1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.295 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
1.778 |
|
|
\[
{}x^{2} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-y^{\prime } x +y = x \left (1-\ln \left (x \right )\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.821 |
|
|
\[
{}x y^{\prime \prime }+2 y^{\prime }+y x = \sec \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.567 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\frac {y}{4} = -\frac {x^{2}}{2}+\frac {1}{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.732 |
|
|
\[
{}\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y = \left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.305 |
|
|
\[
{}\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y = \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
2.379 |
|
|
\[
{}y^{\prime } = x^{2} \left (1+y^{2}\right )
\] |
[_separable] |
✓ |
2.103 |
|
|
\[
{}y^{\prime } = \frac {x^{2}}{1-y^{2}}
\] |
[_separable] |
✓ |
1.005 |
|
|
\[
{}y^{\prime } = \frac {3 x^{2}+4 x +2}{-2+2 y}
\] |
[_separable] |
✓ |
1.457 |
|
|
\[
{}y^{\prime } x -2 \sqrt {y x} = y
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.987 |
|
|
\[
{}y^{\prime } = \frac {x +y-1}{x -y+3}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.657 |
|
|
\[
{}{\mathrm e}^{x}+y+\left (x -2 \sin \left (y\right )\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.918 |
|
|
\[
{}3 x +\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.382 |
|
|
\[
{}y^{2}-y x +x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
1.788 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.449 |
|
|
\[
{}y^{\prime } = \frac {y}{2 x}+\frac {x^{2}}{2 y}
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
1.788 |
|
|
\[
{}y^{\prime } = -\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t}
\] |
[_separable] |
✓ |
2.186 |
|
|
\[
{}y^{\prime } = -\frac {y}{t}-1-y^{2}
\] |
[_rational, _Riccati] |
✓ |
1.154 |
|
|
\[
{}x +y y^{\prime } = a {y^{\prime }}^{2}
\] |
unknown |
✗ |
365.720 |
|
|
\[
{}{y^{\prime }}^{2}-a^{2} y^{2} = 0
\] |
[_quadrature] |
✓ |
0.731 |
|
|
\[
{}{y^{\prime }}^{2} = 4 x^{2}
\] |
[_quadrature] |
✓ |
0.384 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.734 |
|
|
\[
{}s^{\prime \prime }+2 s^{\prime }+s = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.075 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+5 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.372 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }-3 y = 3 x +1
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.932 |
|
|
\[
{}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.018 |
|
|
\[
{}y^{\prime \prime }+y = 4 \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.229 |
|
|
\[
{}y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.098 |
|
|
\[
{}p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u = f \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.782 |
|
|
\[
{}\sin \left (x \right ) u^{\prime \prime }+2 \cos \left (x \right ) u^{\prime }+\sin \left (x \right ) u = 0
\] |
[_Lienard] |
✓ |
3.334 |
|
|
\[
{}3 {y^{\prime \prime }}^{2}-y^{\prime } y^{\prime \prime \prime }-y^{\prime \prime } {y^{\prime }}^{2} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]] |
✓ |
0.481 |
|
|
\[
{}y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1} = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.574 |
|
|
\[
{}x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.082 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y = 4 \,{\mathrm e}^{t}
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.118 |
|
|
\[
{}y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y = 3 \sin \left (t \right )-5 \cos \left (t \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
1.310 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y = g \left (t \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.471 |
|
|
\[
{}y^{\left (5\right )}-\frac {y^{\prime \prime \prime \prime }}{t} = 0
\] |
[[_high_order, _missing_y]] |
✓ |
0.207 |
|
|
\[
{}x x^{\prime \prime }-{x^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.224 |
|
|
\[
{}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-4 y = f \left (x \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
0.611 |
|
|
\[
{}u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.945 |
|
|
\[
{}y^{\prime \prime }+6 y^{\prime }+9 y = 50 \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.968 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+4 y = 50 \,{\mathrm e}^{2 x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.957 |
|
|
\[
{}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.569 |
|
|
\[
{}y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y = 2 \sin \left (3 x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.146 |
|
|
\[
{}y^{\prime \prime }+4 y = x^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.650 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+3 y = x^{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.164 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (3 x +1\right )^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.932 |
|
|
\[
{}y+\sqrt {x^{2}+y^{2}}-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.924 |
|
|
\[
{}{y^{\prime }}^{2} = a^{2}-y^{2}
\] |
[_quadrature] |
✓ |
0.613 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.317 |
|
|
\[
{}y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (x +1\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.813 |
|
|
\[
{}y \left (x^{2} y^{2}+1\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.775 |
|
|
\[
{}2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.491 |
|
|
\[
{}\frac {1}{y}+\sec \left (\frac {y}{x}\right )-\frac {x y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
30.421 |
|
|
\[
{}\phi ^{\prime }-\frac {\phi ^{2}}{2}-\phi \cot \left (\theta \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.855 |
|
|
\[
{}u^{\prime \prime }-\cot \left (\theta \right ) u^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
0.806 |
|
|
\[
{}\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right ) = \frac {\cos \left (2 \theta \right )}{2}+1
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
4.631 |
|
|
\[
{}a y^{\prime \prime } y^{\prime \prime \prime } = \sqrt {1+{y^{\prime \prime }}^{2}}
\] |
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
2.328 |
|
|
\[
{}a^{2} y^{\prime \prime \prime \prime } = y^{\prime \prime }
\] |
[[_high_order, _missing_x]] |
✓ |
0.066 |
|
|
\[
{}y \,{\mathrm e}^{y x}+x \,{\mathrm e}^{y x} y^{\prime } = 0
\] |
[_separable] |
✓ |
1.411 |
|
|
\[
{}x -2 y x +{\mathrm e}^{y}+\left (y-x^{2}+x \,{\mathrm e}^{y}\right ) y^{\prime } = 0
\] |
[_exact] |
✓ |
1.646 |
|
|
\[
{}y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.233 |
|
|
\[
{}\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.636 |
|
|
\[
{}\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (k +1\right ) \eta = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.637 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.344 |
|
|
\[
{}x^{2}-y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
6.296 |
|
|
\[
{}-y+y^{\prime } x = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
1.808 |
|
|
\[
{}-y+y^{\prime } x = x \sqrt {x^{2}-y^{2}}\, y^{\prime }
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.689 |
|
|
\[
{}x +y y^{\prime }+y-y^{\prime } x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.458 |
|