Internal problem ID [9891]
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: 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-f \relax (x ) y^{2}-g^{\prime }\relax (x ) y-a f \relax (x ) {\mathrm e}^{2 g \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 28
dsolve(diff(y(x),x)=f(x)*y(x)^2+diff(g(x),x)*y(x)+a*f(x)*exp(2*g(x)),y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-\sqrt {a}\, \left (\int f \relax (x ) {\mathrm e}^{g \relax (x )}d x \right )+c_{1}\right ) \sqrt {a}\, {\mathrm e}^{g \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.401 (sec). Leaf size: 41
DSolve[y'[x]==f[x]*y[x]^2+g'[x]*y[x]+a*f[x]*Exp[2*g[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {a} e^{g(x)} \tan \left (\sqrt {a} \int _1^xe^{g(K[1])} f(K[1])dK[1]+c_1\right ) \\ \end{align*}