| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
y^{\prime }&=f \left (x \right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.245 |
|
| \begin{align*}
y^{\prime }&=f \left (y\right ) \\
\end{align*} |
[_quadrature] |
✓ |
✓ |
✓ |
✓ |
0.838 |
|
| \begin{align*}
y^{\prime }&=f \left (x \right ) g \left (y\right ) \\
\end{align*} |
[_separable] |
✓ |
✓ |
✓ |
✓ |
1.767 |
|
| \begin{align*}
g \left (x \right ) y^{\prime }&=f_{1} \left (x \right ) y+f_{0} \left (x \right ) \\
\end{align*} |
[_linear] |
✓ |
✓ |
✓ |
✓ |
2.475 |
|
| \begin{align*}
g \left (x \right ) y^{\prime }&=f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n} \\
\end{align*} |
[_Bernoulli] |
✓ |
✓ |
✓ |
✓ |
4.306 |
|
| \begin{align*}
y^{\prime }&=f \left (\frac {y}{x}\right ) \\
\end{align*} |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
✓ |
✓ |
2.737 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.293 |
|
| \begin{align*}
y^{\prime }&=y^{2}-a^{2} x^{2}+3 a \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
3.226 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a^{2} x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.498 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{n} \\
\end{align*} |
[[_Riccati, _special]] |
✓ |
✓ |
✓ |
✗ |
33.462 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
50.579 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,x^{2 n}+c \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.531 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{-n -2} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.412 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
18.212 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
56.734 |
|
| \begin{align*}
y^{\prime }&=\left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
7.660 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
2.771 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
5.169 |
|
| \begin{align*}
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 ) \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.235 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b \,x^{n}+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
77.811 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.671 |
|
| \begin{align*}
x^{4} y^{\prime }&=-y^{2} x^{4}-a^{2} \\
\end{align*} |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
✓ |
✓ |
5.822 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
3.424 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.360 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
85.931 |
|
| \begin{align*}
\left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{-2+n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
9.942 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.371 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b y+c x +k \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
31.214 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+a \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.699 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.484 |
|
| \begin{align*}
y^{\prime }&=y^{2}+\left (\alpha x +\beta \right ) y+a \,x^{2}+b x +c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.419 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.165 |
|
| \begin{align*}
y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
6.713 |
|
| \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] |
✓ |
✗ |
✓ |
✗ |
5.794 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.522 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
10.526 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+b y+c \,x^{2 b} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.380 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+b y+c \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
30.229 |
|
| \begin{align*}
x y^{\prime }&=a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.365 |
|
| \begin{align*}
x y^{\prime }&=x y^{2}+a y+b \,x^{n} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
0.754 |
|
| \begin{align*}
x y^{\prime }+a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}&=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
31.449 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+b y+c \,x^{-n} \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.975 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.649 |
|
| \begin{align*}
x y^{\prime }&=x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
4.739 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{n} y^{2}+b y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.919 |
|
| \begin{align*}
x y^{\prime }&=x^{2 n} a y^{2}+\left (b \,x^{n}-n \right ) y+c \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
6.856 |
|
| \begin{align*}
x y^{\prime }&=a \,x^{2 n +m} y^{2}+\left (b \,x^{n +m}-n \right ) y+c \,x^{m} \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
30.513 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
12.653 |
|
| \begin{align*}
\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 \\
\end{align*} |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
9.631 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
3.052 |
|
| \begin{align*}
2 x^{2} y^{\prime }&=2 y^{2}+3 y x -2 a^{2} x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.559 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \\
\end{align*} |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✓ |
6.107 |
|
| \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] |
✗ |
✓ |
✓ |
✗ |
27.891 |
|
| \begin{align*}
x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{n}+s \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
1.038 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
2.326 |
|
| \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] |
✗ |
✓ |
✓ |
✗ |
27.312 |
|
| \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] |
✗ |
✓ |
✓ |
✗ |
41.385 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 y x +1\right )&=0 \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
4.285 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
459.419 |
|
| \begin{align*}
\left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma &=0 \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
506.861 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.503 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
10.358 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
48.215 |
|
| \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 +c_{2} \lambda \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
51.469 |
|
| \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] |
✓ |
✓ |
✗ |
✗ |
39.024 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
11.657 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
28.248 |
|
| \begin{align*}
x^{3} y^{\prime }&=x^{3} a y^{2}+\left (b \,x^{2}+c \right ) y+s x \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
59.970 |
|
| \begin{align*}
x^{3} y^{\prime }&=x^{3} a y^{2}+x \left (b x +c \right ) y+\alpha x +\beta \\
\end{align*} |
[_rational, _Riccati] |
✗ |
✓ |
✓ |
✗ |
26.872 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
5.309 |
|
| \begin{align*}
x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta &=0 \\
\end{align*} |
[_rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
11.477 |
|
| \begin{align*}
\left (a \,x^{2}+b x +e \right ) \left (x y^{\prime }-y\right )-y^{2}+x^{2}&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
5.822 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
11.872 |
|
| \begin{align*}
x^{n +1} y^{\prime }&=x^{2 n} a y^{2}+b \,x^{n} y+c \,x^{m}+d \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.839 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
39.050 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
13.587 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
1.030 |
|
| \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] |
✓ |
✗ |
✗ |
✗ |
95.883 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
3.641 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) \left (x y^{\prime }-y\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )&=0 \\
\end{align*} |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
✓ |
✗ |
39.348 |
|
| \begin{align*}
y^{\prime }&=a y^{2}+b \,{\mathrm e}^{\lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
23.894 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
4.538 |
|
| \begin{align*}
y^{\prime }&=\sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \\
\end{align*} |
[_Riccati] |
✗ |
✓ |
✓ |
✗ |
7.562 |
|
| \begin{align*}
y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
16.092 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
4.944 |
|
| \begin{align*}
y^{\prime }&=y^{2}+a \,{\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} b -b^{2} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
5.099 |
|
| \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] |
✗ |
✓ |
✗ |
✗ |
34.092 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
2.311 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.807 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✗ |
✓ |
✗ |
42.379 |
|
| \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] |
✓ |
✓ |
✓ |
✓ |
35.550 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
0.696 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.783 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.414 |
|
| \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] |
✗ |
✓ |
✓ |
✗ |
27.215 |
|
| \begin{align*}
y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
2.750 |
|
| \begin{align*}
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} \\
\end{align*} |
[_Riccati] |
✓ |
✓ |
✓ |
✗ |
1.547 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.069 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
26.634 |
|
| \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] |
✓ |
✓ |
✓ |
✗ |
1.185 |
|