|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.751 |
|
|
\[
{}y^{\prime \prime }-\left (a +2 b \,{\mathrm e}^{a x}\right ) y^{\prime }+b^{2} {\mathrm e}^{2 a x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.092 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.634 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+b \,{\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.983 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.636 |
|
|
\[
{}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.699 |
|
|
\[
{}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.793 |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+c \,{\mathrm e}^{\mu x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.716 |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
39.367 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.761 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +c b -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.734 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.691 |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.782 |
|
|
\[
{}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.880 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.835 |
|
|
\[
{}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.973 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right ) y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.383 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{\lambda x} a c +b \mu \,{\mathrm e}^{\mu x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.904 |
|
|
\[
{}\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0
\] |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.637 |
|
|
\[
{}\frac {-2 x^{2}+y^{2}}{x y^{2}-x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y} = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
43.760 |
|
|
\[
{}\frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
18.623 |
|
|
\[
{}y+x +x y^{\prime } = 0
\] |
[_linear] |
✓ |
2.728 |
|
|
\[
{}6 x -2 y+1+\left (2 y-2 x -3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.444 |
|
|
\[
{}\sec \left (x \right ) \cos \left (y\right )^{2}-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
7.178 |
|
|
\[
{}\left (x +1\right ) y^{2}-x^{3} y^{\prime } = 0
\] |
[_separable] |
✓ |
1.888 |
|
|
\[
{}2 \left (1-y^{2}\right ) x y+\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
5.588 |
|
|
\[
{}\sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime } = 0
\] |
[_separable] |
✓ |
3.801 |
|
|
\[
{}x \,{\mathrm e}^{\frac {y}{x}}+y-x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
16.189 |
|
|
\[
{}2 x^{2} y+3 y^{3}-\left (x^{3}+2 x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
43.631 |
|
|
\[
{}y^{2}-x y+x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.414 |
|
|
\[
{}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
73.415 |
|
|
\[
{}y^{3}+x^{3} y^{\prime } = 0
\] |
[_separable] |
✓ |
5.238 |
|
|
\[
{}x +y \cos \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.404 |
|
|
\[
{}4 x +3 y+1+\left (x +y+1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.743 |
|
|
\[
{}4 x -y+2+\left (x +y+3\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
36.918 |
|
|
\[
{}2 x +y-\left (4 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
2.055 |
|
|
\[
{}y+2 x y^{2}-x^{2} y^{3}+2 x^{2} y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.958 |
|
|
\[
{}2 y+3 x y^{2}+\left (x +2 x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
10.855 |
|
|
\[
{}y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
1.915 |
|
|
\[
{}y^{\prime }+y \cot \left (x \right ) = \sec \left (x \right )
\] |
[_linear] |
✓ |
1.838 |
|
|
\[
{}x y^{\prime }+\left (x +1\right ) y = {\mathrm e}^{x}
\] |
[_linear] |
✓ |
1.454 |
|
|
\[
{}y^{\prime }-\frac {2 y}{x +1} = \left (x +1\right )^{3}
\] |
[_linear] |
✓ |
1.619 |
|
|
\[
{}\left (x^{3}+x \right ) y^{\prime }+4 x^{2} y = 2
\] |
[_linear] |
✓ |
1.356 |
|
|
\[
{}x^{2} y^{\prime }+\left (1-2 x \right ) y = x^{2}
\] |
[_linear] |
✓ |
1.832 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }-2 \left (x +1\right ) y = y^{{5}/{2}}
\] |
[_rational, _Bernoulli] |
✓ |
1.944 |
|
|
\[
{}y y^{\prime }+x y^{2} = x
\] |
[_separable] |
✓ |
2.131 |
|
|
\[
{}\sin \left (y\right ) y^{\prime }+\sin \left (x \right ) \cos \left (y\right ) = \sin \left (x \right )
\] |
[_separable] |
✓ |
29.934 |
|
|
\[
{}4 x y^{\prime }+3 y+{\mathrm e}^{x} x^{4} y^{5} = 0
\] |
[_Bernoulli] |
✓ |
2.373 |
|
|
\[
{}y^{\prime }-\frac {y+1}{x +1} = \sqrt {y+1}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
4.238 |
|
|
\[
{}x^{4} y \left (3 y+2 x y^{\prime }\right )+x^{2} \left (4 y+3 x y^{\prime }\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
4.432 |
|
|
\[
{}y^{2} \left (3 y-6 x y^{\prime }\right )-x \left (y-2 x y^{\prime }\right ) = 0
\] |
[_separable] |
✓ |
2.263 |
|
|
\[
{}2 x^{3} y-y^{2}-\left (2 x^{4}+x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
5.063 |
|
|
\[
{}y^{2}-x y+x^{2} y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
2.416 |
|
|
\[
{}\frac {-y+x y^{\prime }}{\sqrt {x^{2}-y^{2}}} = x y^{\prime }
\] |
[‘y=_G(x,y’)‘] |
✓ |
2.016 |
|
|
\[
{}x +y-\left (x -y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
39.520 |
|
|
\[
{}x^{2}+y^{2}-2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
7.570 |
|
|
\[
{}x -y^{2}+2 x y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
2.360 |
|
|
\[
{}-y+x y^{\prime } = x^{2}+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
1.752 |
|
|
\[
{}3 x^{2}+6 x y+3 y^{2}+\left (2 x^{2}+3 x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
41.014 |
|
|
\[
{}2 x +\left (x^{2}+y^{2}+2 y\right ) y^{\prime } = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
1.274 |
|
|
\[
{}y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime } = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
2.426 |
|
|
\[
{}x^{3} y-y^{4}+\left (x y^{3}-x^{4}\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
1.713 |
|
|
\[
{}y^{2}-x^{2}+2 m x y+\left (m y^{2}-m \,x^{2}-2 x y\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
12.822 |
|
|
\[
{}x y^{\prime }-y+2 x^{2} y-x^{3} = 0
\] |
[_linear] |
✓ |
1.710 |
|
|
\[
{}\left (x +y\right ) y^{\prime }-1 = 0
\] |
[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
1.767 |
|
|
\[
{}x +y y^{\prime }+y-x y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
38.083 |
|
|
\[
{}x y^{\prime }-a y+b y^{2} = c \,x^{2 a}
\] |
[_rational, _Riccati] |
✓ |
2.817 |
|
|
\[
{}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
3.731 |
|
|
\[
{}\sqrt {1-y^{2}}+\sqrt {-x^{2}+1}\, y^{\prime } = 0
\] |
[_separable] |
✓ |
40.450 |
|
|
\[
{}y^{\prime }-x^{2} y = x^{5}
\] |
[_linear] |
✓ |
2.175 |
|
|
\[
{}\left (y-x \right )^{2} y^{\prime } = 1
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
3.017 |
|
|
\[
{}x y^{\prime }+y+x^{4} y^{4} {\mathrm e}^{x} = 0
\] |
[_Bernoulli] |
✓ |
3.761 |
|
|
\[
{}\left (1-x \right ) y+\left (1-y\right ) x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.437 |
|
|
\[
{}\left (y-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.166 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
8.071 |
|
|
\[
{}-y+x y^{\prime } = \sqrt {x^{2}-y^{2}}
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
88.928 |
|
|
\[
{}x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
4.102 |
|
|
\[
{}x -2 y+5+\left (2 x -y+4\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.908 |
|
|
\[
{}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{{3}/{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}
\] |
[_linear] |
✓ |
3.993 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime }-x y = a x y^{2}
\] |
[_separable] |
✓ |
2.275 |
|
|
\[
{}x y^{2} \left (3 y+x y^{\prime }\right )-2 y+x y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
5.165 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right )
\] |
[_linear] |
✓ |
2.039 |
|
|
\[
{}5 x y-3 y^{3}+\left (3 x^{2}-7 x y^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
2.607 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2}
\] |
[_linear] |
✓ |
2.404 |
|
|
\[
{}x y^{2}+y-x y^{\prime } = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.391 |
|
|
\[
{}\left (1-x \right ) y-\left (y+1\right ) x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.351 |
|
|
\[
{}3 x^{2} y+\left (x^{3}+x^{3} y^{2}\right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.344 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right ) = \left (x^{2}+y^{2}+x \right ) \left (-y+x y^{\prime }\right )
\] |
[_rational] |
✗ |
3.010 |
|
|
\[
{}2 x +3 y-1+\left (2 x +3 y-5\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.881 |
|
|
\[
{}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
50.793 |
|
|
\[
{}2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
1.645 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )+\sqrt {1+x^{2}+y^{2}}\, \left (y-x y^{\prime }\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.938 |
|
|
\[
{}1+{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.369 |
|
|
\[
{}x y^{\prime }+y-y^{2} \ln \left (x \right ) = 0
\] |
[_Bernoulli] |
✓ |
2.355 |
|
|
\[
{}x^{3} y^{4}+x^{2} y^{3}+x y^{2}+y+\left (x^{4} y^{3}-x^{3} y^{2}-x^{3} y+x \right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
3.082 |
|
|
\[
{}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
24.313 |
|
|
\[
{}{y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+x y = 0
\] |
[_quadrature] |
✓ |
0.491 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }-x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
0.951 |
|
|
\[
{}y^{2}+{y^{\prime }}^{2} = 1
\] |
[_quadrature] |
✓ |
0.646 |
|
|
\[
{}\left (2 x y^{\prime }-y\right )^{2} = 8 x^{3}
\] |
[_linear] |
✓ |
0.581 |
|