|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s = 0
\] |
[_rational, _Riccati] |
✓ |
4.184 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
3.961 |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
3.183 |
|
|
\[
{}\left (a \,x^{n}+b \right ) y^{\prime } = b y^{2}+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✓ |
4.177 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2} = 0
\] |
[_rational, _Riccati] |
✓ |
5.622 |
|
|
\[
{}y^{\prime } = y^{2} a +b y+c x +k
\] |
[_Riccati] |
✓ |
1.475 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+a \,x^{n -1}
\] |
[_Riccati] |
✓ |
2.700 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y+b \,x^{n -1}
\] |
[_Riccati] |
✓ |
3.353 |
|
|
\[
{}y^{\prime } = y^{2}+\left (\alpha x +\beta \right ) y+a \,x^{2}+b x +c
\] |
[_Riccati] |
✓ |
38.499 |
|
|
\[
{}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2}
\] |
[_Riccati] |
✗ |
3.606 |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +m +1}-a \,x^{m}
\] |
[_Riccati] |
✗ |
4.347 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n}
\] |
[_Riccati] |
✗ |
4.779 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1}
\] |
[_Riccati] |
✓ |
7.045 |
|
|
\[
{}y^{\prime } = -a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m}
\] |
[_Riccati] |
✓ |
6.848 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{k -1}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k}
\] |
[_Riccati] |
✗ |
7.751 |
|
|
\[
{}x y^{\prime } = y^{2} a +b y+c \,x^{2 b}
\] |
[_rational, _Riccati] |
✓ |
1.896 |
|
|
\[
{}x y^{\prime } = y^{2} a +b y+c \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
2.133 |
|
|
\[
{}x y^{\prime } = y^{2} a +\left (n +b \,x^{n}\right ) y+c \,x^{2 n}
\] |
[_rational, _Riccati] |
✓ |
3.258 |
|
|
\[
{}x y^{\prime } = x y^{2}+a y+b \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
2.068 |
|
|
\[
{}x y^{\prime }+a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
5.704 |
|
|
\[
{}x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{-n}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.760 |
|
|
\[
{}x y^{\prime } = a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m}
\] |
[_rational, _Riccati] |
✓ |
2.371 |
|
|
\[
{}x y^{\prime } = x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m}
\] |
[_rational, _Riccati] |
✓ |
2.354 |
|
|
\[
{}x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
2.683 |
|
|
\[
{}x y^{\prime } = a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c
\] |
[_rational, _Riccati] |
✓ |
3.693 |
|
|
\[
{}x y^{\prime } = a \,x^{2 n +m} y^{2}+\left (b \,x^{m +n}-n \right ) y+c \,x^{m}
\] |
[_rational, _Riccati] |
✓ |
36.730 |
|
|
\[
{}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
18.160 |
|
|
\[
{}\left (a x +c \right ) y^{\prime } = \alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
3.281 |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
1.525 |
|
|
\[
{}2 x^{2} y^{\prime } = 2 y^{2}+3 x y-2 a^{2} x
\] |
[_rational, _Riccati] |
✓ |
1.822 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.365 |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma
\] |
[_rational, _Riccati] |
✓ |
6.519 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{n}+s
\] |
[_rational, _Riccati] |
✓ |
2.732 |
|
|
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n}
\] |
[_rational, _Riccati] |
✓ |
3.829 |
|
|
\[
{}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma
\] |
[_rational, _Riccati] |
✓ |
8.530 |
|
|
\[
{}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2}
\] |
[_rational, _Riccati] |
✗ |
537.773 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right ) = 0
\] |
[_rational, _Riccati] |
✗ |
6.538 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0
\] |
[_rational, _Riccati] |
✓ |
400.595 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0
\] |
[_rational, _Riccati] |
✓ |
427.681 |
|
|
\[
{}\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 x y+\left (-a +1\right ) x^{2}-b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.538 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
4.771 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c
\] |
[_rational, _Riccati] |
✗ |
457.832 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}
\] |
[_rational, _Riccati] |
✓ |
51.717 |
|
|
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0}
\] |
[_rational, _Riccati] |
✓ |
44.242 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.951 |
|
|
\[
{}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0
\] |
[_rational, _Riccati] |
✓ |
18.652 |
|
|
\[
{}x^{3} y^{\prime } = a \,x^{3} y^{2}+\left (b \,x^{2}+c \right ) y+s x
\] |
[_rational, _Riccati] |
✓ |
10.506 |
|
|
\[
{}x^{3} y^{\prime } = a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta
\] |
[_rational, _Riccati] |
✓ |
4.954 |
|
|
\[
{}x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x = 0
\] |
[_rational, _Riccati] |
✓ |
4.924 |
|
|
\[
{}x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0
\] |
[_rational, _Riccati] |
✓ |
5.718 |
|
|
\[
{}\left (a \,x^{2}+b x +e \right ) \left (-y+x y^{\prime }\right )-y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.751 |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s = 0
\] |
[_rational, _Riccati] |
✓ |
7.302 |
|
|
\[
{}a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0
\] |
[_rational, _Riccati] |
✗ |
5.204 |
|
|
\[
{}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d
\] |
[_Riccati] |
✓ |
3.859 |
|
|
\[
{}x \left (a \,x^{k}+b \right ) y^{\prime } = \alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n}
\] |
[_rational, _Riccati] |
✓ |
40.350 |
|
|
\[
{}x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0
\] |
[_rational, _Riccati] |
✓ |
21.321 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✗ |
93.551 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c
\] |
[_rational, _Riccati] |
✗ |
83.558 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}
\] |
[_rational, _Riccati] |
✗ |
129.325 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+x y^{\prime }\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right ) = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
40.727 |
|
|
\[
{}y^{\prime } = y^{2} a +b \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
1.475 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
2.096 |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
3.280 |
|
|
\[
{}y^{\prime } = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c
\] |
[_Riccati] |
✓ |
2.024 |
|
|
\[
{}y^{\prime } = y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
2.599 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2}
\] |
[_Riccati] |
✓ |
2.166 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4}
\] |
[_Riccati] |
✓ |
5.774 |
|
|
\[
{}y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2}
\] |
[_Riccati] |
✓ |
7.751 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x}
\] |
[_Riccati] |
✓ |
2.647 |
|
|
\[
{}y^{\prime } = b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✗ |
4.508 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
2.407 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}
\] |
[_Riccati] |
✓ |
2.276 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
3.358 |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}
\] |
[_Riccati] |
✓ |
3.280 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
421.163 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}
\] |
[_Riccati] |
✓ |
3.651 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y+c \,{\mathrm e}^{\mu x}
\] |
[_Riccati] |
✓ |
3.377 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (2 n +1\right ) x}
\] |
[_Riccati] |
✗ |
5.005 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.524 |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}
\] |
[_Riccati] |
✗ |
73.860 |
|
|
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x} = 0
\] |
[_Riccati] |
✓ |
4.616 |
|
|
\[
{}y^{\prime } = y^{2}+a x \,{\mathrm e}^{\lambda x} y+a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
2.456 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x}
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
2.171 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n}
\] |
[_Riccati] |
✗ |
6.925 |
|
|
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x}
\] |
[_Riccati] |
✗ |
4.819 |
|
|
\[
{}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n}
\] |
[_Riccati] |
✓ |
3.177 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1}
\] |
[_Riccati] |
✓ |
4.330 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✗ |
6.171 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}
\] |
[_Riccati] |
✓ |
3.053 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
5.936 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.751 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n}
\] |
[_Riccati] |
✗ |
6.789 |
|
|
\[
{}y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n}
\] |
[_Riccati] |
✓ |
5.678 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
3.869 |
|
|
\[
{}x y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
3.197 |
|
|
\[
{}x y^{\prime } = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✗ |
19.308 |
|
|
\[
{}y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}}
\] |
[_Riccati] |
✗ |
2.985 |
|
|
\[
{}y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+b^{2} a
\] |
[_Riccati] |
✓ |
2.051 |
|
|
\[
{}y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}
\] |
[_Riccati] |
✓ |
3.227 |
|
|
\[
{}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}}
\] |
[_Riccati] |
✓ |
3.958 |
|