Internal problem ID [9865]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary
functions (but not containing their derivatives).
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-f \relax (x ) y^{2}-\lambda y-a^{2} {\mathrm e}^{2 \lambda x} f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 29
dsolve(diff(y(x),x)=f(x)*y(x)^2+lambda*y(x)+a^2*exp(2*lambda*x)*f(x),y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-a \left (\int {\mathrm e}^{\lambda x} f \relax (x )d x \right )+c_{1}\right ) a \,{\mathrm e}^{\lambda x} \]
✓ Solution by Mathematica
Time used: 0.396 (sec). Leaf size: 47
DSolve[y'[x]==f[x]*y[x]^2+\[Lambda]*y[x]+a^2*Exp[2*\[Lambda]*x]*f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {a^2} e^{\lambda x} \tan \left (\sqrt {a^2} \int _1^xe^{\lambda K[1]} f(K[1])dK[1]+c_1\right ) \\ \end{align*}