Internal problem ID [10618]
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]
\[ \boxed {y^{\prime }-y^{2} f \left (x \right )-y \lambda =a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.046 (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)
\[ y \left (x \right ) = -\tan \left (-a \left (\int f \left (x \right ) {\mathrm e}^{\lambda x}d x \right )+c_{1} \right ) a \,{\mathrm e}^{\lambda x} \]
✓ Solution by Mathematica
Time used: 0.615 (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]
\[ y(x)\to \sqrt {a^2} e^{\lambda x} \tan \left (\sqrt {a^2} \int _1^xe^{\lambda K[1]} f(K[1])dK[1]+c_1\right ) \]