Internal problem ID [10634]
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]
\[ \boxed {y^{\prime }-f \left (x \right ) y^{2}-g^{\prime }\left (x \right ) y=a f \left (x \right ) {\mathrm e}^{2 g \left (x \right )}} \]
✓ Solution by Maple
Time used: 0.031 (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)
\[ y \left (x \right ) = -\tan \left (-\sqrt {a}\, \left (\int f \left (x \right ) {\mathrm e}^{g \left (x \right )}d x \right )+c_{1} \right ) \sqrt {a}\, {\mathrm e}^{g \left (x \right )} \]
✓ Solution by Mathematica
Time used: 0.635 (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]
\[ y(x)\to \sqrt {a} e^{g(x)} \tan \left (\sqrt {a} \int _1^xe^{g(K[1])} f(K[1])dK[1]+c_1\right ) \]