|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = f \left (x \right )
\] |
[_quadrature] |
✓ |
0.413 |
|
|
\[
{}y^{\prime } = f \left (y\right )
\] |
[_quadrature] |
✓ |
0.589 |
|
|
\[
{}y^{\prime } = f \left (x \right ) g \left (y\right )
\] |
[_separable] |
✓ |
0.957 |
|
|
\[
{}g \left (x \right ) y^{\prime } = f_{1} \left (x \right ) y+f_{0} \left (x \right )
\] |
[_linear] |
✓ |
2.022 |
|
|
\[
{}g \left (x \right ) y^{\prime } = f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n}
\] |
[_Bernoulli] |
✓ |
2.375 |
|
|
\[
{}y^{\prime } = f \left (\frac {y}{x}\right )
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
1.222 |
|
|
\[
{}y^{\prime } = a y^{2}+b x +c
\] |
[_Riccati] |
✓ |
1.256 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2} x^{2}+3 a
\] |
[_Riccati] |
✓ |
1.755 |
|
|
\[
{}y^{\prime } = y^{2}+a^{2} x^{2}+b x +c
\] |
[_Riccati] |
✓ |
9.671 |
|
|
\[
{}y^{\prime } = a y^{2}+b \,x^{n}
\] |
[[_Riccati, _special]] |
✓ |
1.557 |
|
|
\[
{}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n}
\] |
[_Riccati] |
✗ |
605.206 |
|
|
\[
{}y^{\prime } = a y^{2}+b \,x^{2 n}+c \,x^{n -1}
\] |
[_Riccati] |
✓ |
3.178 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{-n -2}
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
2.460 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m}
\] |
[_Riccati] |
✓ |
1.957 |
|
|
\[
{}y^{\prime } = y^{2}+k \left (a x +b \right )^{n} \left (c x +d \right )^{-n -4}
\] |
[_Riccati] |
✗ |
7.435 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m}
\] |
[_Riccati] |
✗ |
419.558 |
|
|
\[
{}y^{\prime } = \left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c
\] |
[_Riccati] |
✓ |
51.216 |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
3.377 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b
\] |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
1.824 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right )
\] |
[_rational, _Riccati] |
✓ |
2.404 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b \,x^{n}+c
\] |
[_rational, _Riccati] |
✓ |
2.180 |
|
|
\[
{}x^{2} y^{\prime } = x^{2} y^{2}+a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4}
\] |
[_Riccati] |
✗ |
3.985 |
|
|
\[
{}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
8.285 |
|
|
\[
{}x^{4} y^{\prime } = -y^{2} x^{4}-a^{2}
\] |
[_rational, [_Riccati, _special]] |
✓ |
2.924 |
|
|
\[
{}a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s = 0
\] |
[_rational, _Riccati] |
✓ |
4.432 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
3.630 |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
2.947 |
|
|
\[
{}\left (a \,x^{n}+b \right ) y^{\prime } = b y^{2}+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✓ |
3.724 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2} = 0
\] |
[_rational, _Riccati] |
✓ |
5.722 |
|
|
\[
{}y^{\prime } = a y^{2}+b y+c x +k
\] |
[_Riccati] |
✓ |
1.556 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+a \,x^{n -1}
\] |
[_Riccati] |
✓ |
2.720 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+b \,x^{n -1}
\] |
[_Riccati] |
✓ |
3.468 |
|
|
\[
{}y^{\prime } = y^{2}+\left (\alpha x +\beta \right ) y+a \,x^{2}+b x +c
\] |
[_Riccati] |
✓ |
37.232 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2}
\] |
[_Riccati] |
✗ |
3.591 |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +m +1}-a \,x^{m}
\] |
[_Riccati] |
✗ |
4.134 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n}
\] |
[_Riccati] |
✗ |
4.861 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1}
\] |
[_Riccati] |
✓ |
6.709 |
|
|
\[
{}y^{\prime } = -a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m}
\] |
[_Riccati] |
✓ |
6.530 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{k -1}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k}
\] |
[_Riccati] |
✗ |
7.672 |
|
|
\[
{}y^{\prime } x = a y^{2}+b y+c \,x^{2 b}
\] |
[_rational, _Riccati] |
✓ |
2.749 |
|
|
\[
{}y^{\prime } x = a y^{2}+b y+c \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
2.194 |
|
|
\[
{}y^{\prime } x = a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n}
\] |
[_rational, _Riccati] |
✓ |
3.087 |
|
|
\[
{}y^{\prime } x = x y^{2}+a y+b \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
2.071 |
|
|
\[
{}y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
5.671 |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b y+c \,x^{-n}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.651 |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m}
\] |
[_rational, _Riccati] |
✓ |
2.249 |
|
|
\[
{}y^{\prime } x = x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m}
\] |
[_rational, _Riccati] |
✓ |
2.253 |
|
|
\[
{}y^{\prime } x = a \,x^{n} y^{2}+b y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
2.670 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c
\] |
[_rational, _Riccati] |
✓ |
3.631 |
|
|
\[
{}y^{\prime } x = a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
51.775 |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
19.092 |
|
|
\[
{}\left (a x +c \right ) y^{\prime } = \alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
3.351 |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
1.589 |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+3 x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
1.860 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.427 |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma
\] |
[_rational, _Riccati] |
✓ |
6.372 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{n}+s
\] |
[_rational, _Riccati] |
✓ |
2.515 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
3.553 |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma
\] |
[_rational, _Riccati] |
✓ |
7.751 |
|
|
\[
{}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2}
\] |
[_rational, _Riccati] |
✗ |
530.356 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right ) = 0
\] |
[_rational, _Riccati] |
✗ |
6.318 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0
\] |
[_rational, _Riccati] |
✓ |
366.736 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0
\] |
[_rational, _Riccati] |
✓ |
431.310 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 x y+\left (1-a \right ) x^{2}-b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.534 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
4.789 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c
\] |
[_rational, _Riccati] |
✗ |
460.123 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}
\] |
[_rational, _Riccati] |
✓ |
50.296 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0}
\] |
[_rational, _Riccati] |
✓ |
45.411 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.959 |
|
|
\[
{}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
18.750 |
|
|
\[
{}x^{3} y^{\prime } = a \,x^{3} y^{2}+\left (b \,x^{2}+c \right ) y+s x
\] |
[_rational, _Riccati] |
✓ |
10.247 |
|
|
\[
{}x^{3} y^{\prime } = a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta
\] |
[_rational, _Riccati] |
✓ |
4.680 |
|
|
\[
{}x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x = 0
\] |
[_rational, _Riccati] |
✓ |
4.776 |
|
|
\[
{}x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0
\] |
[_rational, _Riccati] |
✓ |
5.490 |
|
|
\[
{}\left (a \,x^{2}+b x +e \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.599 |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s = 0
\] |
[_rational, _Riccati] |
✓ |
6.966 |
|
|
\[
{}a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0
\] |
[_rational, _Riccati] |
✗ |
5.078 |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
3.657 |
|
|
\[
{}x \left (a \,x^{k}+b \right ) y^{\prime } = \alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n}
\] |
[_rational, _Riccati] |
✓ |
39.628 |
|
|
\[
{}x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0
\] |
[_rational, _Riccati] |
✓ |
20.571 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✗ |
97.294 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c
\] |
[_rational, _Riccati] |
✗ |
85.003 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}
\] |
[_rational, _Riccati] |
✗ |
130.342 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+y^{\prime } x \right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right ) = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
46.423 |
|
|
\[
{}y^{\prime } = a y^{2}+b \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
1.405 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
1.981 |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
3.180 |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c
\] |
[_Riccati] |
✓ |
1.931 |
|
|
\[
{}y^{\prime } = y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
2.586 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2}
\] |
[_Riccati] |
✓ |
2.156 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\] |
[_Riccati] |
✓ |
5.592 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2}
\] |
[_Riccati] |
✓ |
7.508 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x}
\] |
[_Riccati] |
✓ |
2.588 |
|
|
\[
{}y^{\prime } = b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✗ |
4.007 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
2.376 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✓ |
2.161 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
3.265 |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
3.209 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
416.536 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}
\] |
[_Riccati] |
✓ |
3.484 |
|