|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = \frac {1}{x +2 y+1}
\] |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
1.752 |
|
|
\[
{}y^{\prime } = -\frac {x +y}{3 x +3 y-4}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.319 |
|
|
\[
{}y^{\prime } = \tan \left (x \right ) \cos \left (y\right ) \left (\cos \left (y\right )+\sin \left (y\right )\right )
\] |
[_separable] |
✓ |
4.855 |
|
|
\[
{}x \left (1-2 x^{2} y\right ) y^{\prime }+y = 3 x^{2} y^{2}
\] |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.194 |
|
|
\[
{}y^{\prime }+\frac {x y}{a^{2}+x^{2}} = x
\] |
[_linear] |
✓ |
5.631 |
|
|
\[
{}y^{\prime } = \frac {4 y^{2}}{x^{2}}-y^{2}
\] |
[_separable] |
✓ |
3.053 |
|
|
\[
{}y^{\prime }-\frac {y}{x} = 1
\] |
[_linear] |
✓ |
2.336 |
|
|
\[
{}y^{\prime }-y \tan \left (x \right ) = 1
\] |
[_linear] |
✓ |
1.708 |
|
|
\[
{}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.793 |
|
|
\[
{}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.729 |
|
|
\[
{}y^{\prime } \sin \left (x \right )+2 y \cos \left (x \right ) = 1
\] |
[_linear] |
✓ |
3.264 |
|
|
\[
{}\left (5 x +y-7\right ) y^{\prime } = 3 x +3 y+3
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
7.307 |
|
|
\[
{}y^{\prime } x +y-\frac {y^{2}}{x^{{3}/{2}}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
7.154 |
|
|
\[
{}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
11.348 |
|
|
\[
{}\left (2 \sin \left (y\right )-x \right ) y^{\prime } = \tan \left (y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
12.224 |
|
|
\[
{}y^{\prime \prime }+{y^{\prime }}^{2}+y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.262 |
|
|
\[
{}x^{\prime \prime }+\omega _{0}^{2} x = a \cos \left (\omega t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.998 |
|
|
\[
{}f^{\prime \prime }+2 f^{\prime }+5 f = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.115 |
|
|
\[
{}f^{\prime \prime }+2 f^{\prime }+5 f = {\mathrm e}^{-t} \cos \left (3 t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
11.724 |
|
|
\[
{}f^{\prime \prime }+6 f^{\prime }+9 f = {\mathrm e}^{-t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.188 |
|
|
\[
{}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.541 |
|
|
\[
{}f^{\prime \prime }+8 f^{\prime }+12 f = 12 \,{\mathrm e}^{-4 t}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.336 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+y = 4 \,{\mathrm e}^{-x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.160 |
|
|
\[
{}y^{\prime \prime \prime }-12 y^{\prime }+16 y = 32 x -8
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.787 |
|
|
\[
{}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y} = 2 a^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.532 |
|
|
\[
{}x^{2} y^{\prime \prime }-y^{\prime } x +y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.038 |
|
|
\[
{}\left (x +1\right )^{2} y^{\prime \prime }+3 \left (x +1\right ) y^{\prime }+y = x^{2}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
3.924 |
|
|
\[
{}\left (x -2\right ) y^{\prime \prime }+3 y^{\prime }+\frac {4 y}{x^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.059 |
|
|
\[
{}y^{\prime \prime }-y = x^{n}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.862 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+y = 2 x \,{\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.970 |
|
|
\[
{}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right )
\] |
[[_3rd_order, _exact, _nonlinear]] |
✗ |
0.055 |
|
|
\[
{}x y^{\prime \prime \prime }+2 y^{\prime \prime } = A x
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.253 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+6\right ) y = {\mathrm e}^{-x^{2}} \sin \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
8.064 |
|
|
\[
{}\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y = 0
\] |
[_Gegenbauer] |
✓ |
1.166 |
|
|
\[
{}4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.210 |
|
|
\[
{}z y^{\prime \prime }-2 y^{\prime }+9 z^{5} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.147 |
|
|
\[
{}f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.374 |
|
|
\[
{}z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (1+z \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.856 |
|
|
\[
{}z y^{\prime \prime }-2 y^{\prime }+y z = 0
\] |
[_Lienard] |
✓ |
1.115 |
|
|
\[
{}y^{\prime \prime }-2 z y^{\prime }-2 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.007 |
|
|
\[
{}z \left (1-z \right ) y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0
\] |
[_Jacobi] |
✓ |
1.408 |
|
|
\[
{}z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.145 |
|
|
\[
{}\left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.072 |
|
|
\[
{}\left (z^{2}+5 z +7\right ) y^{\prime \prime }+2 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.372 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{z^{3}} = 0
\] |
[[_Emden, _Fowler]] |
✗ |
0.053 |
|
|
\[
{}z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y = 0
\] |
[_Laguerre] |
✓ |
1.323 |
|
|
\[
{}\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.122 |
|
|
\[
{}y^{\prime } = 2 y x
\] |
[_separable] |
✓ |
1.595 |
|
|
\[
{}y^{\prime } = \frac {y^{2}}{x^{2}+1}
\] |
[_separable] |
✓ |
3.218 |
|
|
\[
{}{\mathrm e}^{x +y} y^{\prime }-1 = 0
\] |
[_separable] |
✓ |
3.300 |
|
|
\[
{}y^{\prime } = \frac {y}{x \ln \left (x \right )}
\] |
[_separable] |
✓ |
2.565 |
|
|
\[
{}y-\left (x -2\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
3.161 |
|
|
\[
{}y^{\prime } = \frac {2 x \left (-1+y\right )}{x^{2}+3}
\] |
[_separable] |
✓ |
1.586 |
|
|
\[
{}y-y^{\prime } x = 3-2 x^{2} y^{\prime }
\] |
[_separable] |
✓ |
2.393 |
|
|
\[
{}y^{\prime } = \frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}-1
\] |
[_separable] |
✓ |
5.289 |
|
|
\[
{}y^{\prime } = \frac {x \left (-1+y^{2}\right )}{2 \left (x -2\right ) \left (x -1\right )}
\] |
[_separable] |
✓ |
4.128 |
|
|
\[
{}y^{\prime } = \frac {x^{2} y-32}{-x^{2}+16}+32
\] |
[_linear] |
✓ |
1.951 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }-y+c = 0
\] |
[_separable] |
✓ |
4.441 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y^{2} = -1
\] |
[_separable] |
✓ |
4.049 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }+y x = a x
\] |
[_separable] |
✓ |
4.175 |
|
|
\[
{}y^{\prime } = 1-\frac {\sin \left (x +y\right )}{\sin \left (y\right ) \cos \left (x \right )}
\] |
[_separable] |
✓ |
5.864 |
|
|
\[
{}y^{\prime } = y^{3} \sin \left (x \right )
\] |
[_separable] |
✓ |
4.183 |
|
|
\[
{}y^{\prime }-y = {\mathrm e}^{2 x}
\] |
[[_linear, ‘class A‘]] |
✓ |
2.067 |
|
|
\[
{}x^{2} y^{\prime }-4 y x = x^{7} \sin \left (x \right )
\] |
[_linear] |
✓ |
2.681 |
|
|
\[
{}y^{\prime }+2 y x = 2 x^{3}
\] |
[_linear] |
✓ |
2.312 |
|
|
\[
{}y^{\prime }+\frac {2 x y}{x^{2}+1} = 4 x
\] |
[_linear] |
✓ |
2.683 |
|
|
\[
{}y^{\prime }+\frac {2 x y}{x^{2}+1} = \frac {4}{\left (x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
2.682 |
|
|
\[
{}2 \cos \left (x \right )^{2} y^{\prime }+y \sin \left (2 x \right ) = 4 \cos \left (x \right )^{4}
\] |
[_linear] |
✓ |
5.435 |
|
|
\[
{}y^{\prime }+\frac {y}{x \ln \left (x \right )} = 9 x^{2}
\] |
[_linear] |
✓ |
2.684 |
|
|
\[
{}y^{\prime }-y \tan \left (x \right ) = 8 \sin \left (x \right )^{3}
\] |
[_linear] |
✓ |
3.244 |
|
|
\[
{}t x^{\prime }+2 x = 4 \,{\mathrm e}^{t}
\] |
[_linear] |
✓ |
2.375 |
|
|
\[
{}y^{\prime } = \sin \left (x \right ) \left (y \sec \left (x \right )-2\right )
\] |
[_linear] |
✓ |
3.168 |
|
|
\[
{}1-y \sin \left (x \right )-\cos \left (x \right ) y^{\prime } = 0
\] |
[_linear] |
✓ |
3.308 |
|
|
\[
{}y^{\prime }-\frac {y}{x} = 2 x^{2} \ln \left (x \right )
\] |
[_linear] |
✓ |
2.227 |
|
|
\[
{}y^{\prime }+\alpha y = {\mathrm e}^{\beta x}
\] |
[[_linear, ‘class A‘]] |
✓ |
2.578 |
|
|
\[
{}y^{\prime }+\frac {m}{x} = \ln \left (x \right )
\] |
[_quadrature] |
✓ |
0.172 |
|
|
\[
{}\left (3 x -y\right ) y^{\prime } = 3 y
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
5.603 |
|
|
\[
{}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
3.928 |
|
|
\[
{}\sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right ) = x \cos \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
14.253 |
|
|
\[
{}y^{\prime } x = \sqrt {16 x^{2}-y^{2}}+y
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
129.599 |
|
|
\[
{}y^{\prime } x -y = \sqrt {9 x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
11.467 |
|
|
\[
{}x \left (x^{2}-y^{2}\right )-x \left (x^{2}+y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.234 |
|
|
\[
{}y^{\prime } x +y \ln \left (x \right ) = y \ln \left (y\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
8.645 |
|
|
\[
{}y^{\prime } = \frac {y^{2}+2 y x -2 x^{2}}{x^{2}-y x +y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
19.121 |
|
|
\[
{}2 x y y^{\prime }-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}-2 y^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
6.312 |
|
|
\[
{}x^{2} y^{\prime } = y^{2}+3 y x +x^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
2.789 |
|
|
\[
{}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.760 |
|
|
\[
{}2 x \left (y+2 x \right ) y^{\prime } = y \left (4 x -y\right )
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
6.592 |
|
|
\[
{}y^{\prime } x = x \tan \left (\frac {y}{x}\right )+y
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
7.882 |
|
|
\[
{}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{y x}
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
68.953 |
|
|
\[
{}y^{\prime \prime }-25 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.258 |
|
|
\[
{}y^{\prime \prime }+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.002 |
|
|
\[
{}y^{\prime \prime }+y^{\prime }-2 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.269 |
|
|
\[
{}y^{\prime } = -y^{2}
\] |
[_quadrature] |
✓ |
0.257 |
|
|
\[
{}y^{\prime } = \frac {y}{2 x}
\] |
[_separable] |
✓ |
2.881 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime }+5 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.428 |
|
|
\[
{}y^{\prime \prime }-9 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
3.187 |
|
|
\[
{}x^{2} y^{\prime \prime }+5 y^{\prime } x +3 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
2.122 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.823 |
|
|
\[
{}x^{2} y^{\prime \prime }-3 y^{\prime } x +13 y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
4.223 |
|