61.20.4 problem 37

Internal problem ID [12306]
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
Date solved : Tuesday, January 28, 2025 at 02:34:37 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=g \left (x \right ) \left (y-f \left (x \right )\right )^{2}+f^{\prime }\left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 17

dsolve(diff(y(x),x)=g(x)*(y(x)-f(x))^2+diff(f(x),x),y(x), singsol=all)
 

\[ y = f+\frac {1}{c_{1} -\int g \left (x \right )d x} \]

✓ Solution by Mathematica

Time used: 0.219 (sec). Leaf size: 31

DSolve[D[y[x],x]==g[x]*(y[x]-f[x])^2+D[ f[x],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*}