| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=f \left (x \right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.160 |
|
| \begin{align*}
y^{\prime }&=f \left (y\right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.389 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) g \left (y\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
1.208 |
|
| \begin{align*}
g \left (x \right ) y^{\prime }&=f_{1} \left (x \right ) y+f_{0} \left (x \right ) \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
1.704 |
|
| \begin{align*}
g \left (x \right ) y^{\prime }&=f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n} \\
\end{align*} |
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
2.516 |
|
| \begin{align*}
y^{\prime }&=f \left (\frac {y}{x}\right ) \\
\end{align*} |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
✓ |
✓ |
1.512 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.160 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2} x^{2}+3 a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.000 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a^{2} x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.776 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{n} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
28.630 |
|
| \begin{align*}
y^{\prime }&=y^{2}+x^{n -1} a n -a^{2} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
53.692 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{2 n}+c \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.451 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{-n -2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.618 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
39.188 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
60.329 |
|
| \begin{align*}
y^{\prime }&=\left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.905 |
|
| \begin{align*}
\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.369 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
3.692 |
|
| \begin{align*}
x^{2} y^{\prime }&=y^{2} x^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.025 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \,x^{n}+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
81.127 |
|
| \begin{align*}
\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
10.352 |
|
| \begin{align*}
x^{4} y^{\prime }&=-y^{2} x^{4}-a^{2} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
5.801 |
|
| \begin{align*}
a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✗ |
✗ |
1.885 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.336 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
80.228 |
|
| \begin{align*}
\left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.027 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{-2+n}+b m \left (m -1\right ) x^{m -2}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.655 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b y+c x +k \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
31.329 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+a \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.048 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.164 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\left (x \alpha +\beta \right ) y+a \,x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.486 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.505 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+x^{m} b c -a \,c^{2} x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.171 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
4.290 |
|
| \begin{align*}
y^{\prime }&=-a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.887 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{-1+k}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✗ |
✗ |
6.929 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b y+c \,x^{2 b} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.401 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+b y+c \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
31.400 |
|
| \begin{align*}
y^{\prime } x&=a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.117 |
|
| \begin{align*}
y^{\prime } x&=x y^{2}+a y+b \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.376 |
|
| \begin{align*}
y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
32.487 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b y+c \,x^{-n} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.801 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.622 |
|
| \begin{align*}
y^{\prime } x&=x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.651 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{n} y^{2}+b y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.880 |
|
| \begin{align*}
y^{\prime } x&=x^{2 n} a y^{2}+\left (b \,x^{n}-n \right ) y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.948 |
|
| \begin{align*}
y^{\prime } x&=a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
32.151 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
13.003 |
|
| \begin{align*}
\left (a x +c \right ) y^{\prime }&=\alpha \left (a y+b x \right )^{2}+\beta \left (a y+b x \right )-b x +\gamma \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.968 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.692 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+3 y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.902 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
3.059 |
|
| \begin{align*}
x^{2} y^{\prime }&=c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
30.799 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{n}+s \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.503 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.203 |
|
| \begin{align*}
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 \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
25.765 |
|
| \begin{align*}
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} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
13.326 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime }+\lambda \left (1-2 y x +y^{2}\right )&=0 \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
1.505 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha }&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
463.421 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma &=0 \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
538.798 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+y^{2}-2 y x +\left (1-a \right ) x^{2}-b&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.536 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.842 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
73.844 |
|
| \begin{align*}
\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} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
112.030 |
|
| \begin{align*}
\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} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✗ |
✗ |
77.736 |
|
| \begin{align*}
\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2}&=0 \\
\end{align*} |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✓ |
8.698 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
25.053 |
|
| \begin{align*}
x^{3} y^{\prime }&=a \,x^{3} y^{2}+\left (b \,x^{2}+c \right ) y+s x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
61.177 |
|
| \begin{align*}
x^{3} y^{\prime }&=a \,x^{3} y^{2}+x \left (b x +c \right ) y+x \alpha +\beta \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
24.248 |
|
| \begin{align*}
x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.826 |
|
| \begin{align*}
x^{2} \left (a +x \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+x \alpha +\beta &=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.714 |
|
| \begin{align*}
\left (a \,x^{2}+b x +e \right ) \left (-y+y^{\prime } x \right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.325 |
|
| \begin{align*}
x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
7.924 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+b \,x^{n} y+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.915 |
|
| \begin{align*}
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} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
43.566 |
|
| \begin{align*}
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 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
8.921 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=c y^{2}-b \,x^{m -1} y+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✗ |
✗ |
✗ |
1.215 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=a \,x^{-2+n} y^{2}+b \,x^{m -1} y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✗ |
✗ |
✗ |
89.496 |
|
| \begin{align*}
\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} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
2.227 |
|
| \begin{align*}
\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 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
37.729 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.864 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.908 |
|
| \begin{align*}
y^{\prime }&=\sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
4.749 |
|
| \begin{align*}
y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
29.149 |
|
| \begin{align*}
y^{\prime }&=y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.327 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.041 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✗ |
✗ |
40.002 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.240 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{x k} y^{2}+b \,{\mathrm e}^{s x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.848 |
|
| \begin{align*}
y^{\prime }&=b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
90.762 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x} \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
✓ |
✓ |
67.860 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (2 \lambda +\mu \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.415 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.894 |
|
| \begin{align*}
y^{\prime }&=-\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.684 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
29.698 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{x k} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-x k} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.240 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} y^{2}+\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y+c \,{\mathrm e}^{\mu x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.948 |
|
| \begin{align*}
y^{\prime }&={\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.644 |
|
| \begin{align*}
\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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
22.509 |
|
| \begin{align*}
\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 \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.612 |
|