Internal problem ID [9887]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing arbitrary
functions and their derivatives.
Problem number: 37.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-g \relax (x ) \left (y-f \relax (x )\right )^{2}-f^{\prime }\relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 17
dsolve(diff(y(x),x)=g(x)*(y(x)-f(x))^2+diff(f(x),x),y(x), singsol=all)
\[ y \relax (x ) = f \relax (x )+\frac {1}{c_{1}-\left (\int g \relax (x )d x \right )} \]
✓ Solution by Mathematica
Time used: 0.224 (sec). Leaf size: 31
DSolve[y'[x]==g[x]*(y[x]-f[x])^2+f'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to f(x)+\frac {1}{-\int _1^xg(K[2])dK[2]+c_1} \\ y(x)\to f(x) \\ \end{align*}