61.19.15 problem 15

Internal problem ID [12284]
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
Date solved : Tuesday, January 28, 2025 at 02:09:05 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=f \left (x \right ) y^{2}+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.117 (sec). Leaf size: 26

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)
 

\[ y = -\tan \left (-a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )+c_{1} \right ) a \,{\mathrm e}^{\lambda x} \]

✓ Solution by Mathematica

Time used: 0.393 (sec). Leaf size: 47

DSolve[D[y[x],x]==f[x]*y[x]^2+\[Lambda]*y[x]+a^2*Exp[2*\[Lambda]*x]*f[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {a^2} e^{\lambda x} \tan \left (\sqrt {a^2} \int _1^xe^{\lambda K[1]} f(K[1])dK[1]+c_1\right ) \]