61.19.13 problem 13

Internal problem ID [12282]
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 : 13
Date solved : Tuesday, January 28, 2025 at 02:08:54 AM
CAS classification : [_Riccati]

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

✗ Solution by Maple

dsolve(diff(y(x),x)=f(x)*y(x)^2-a*exp(lambda*x)*f(x)*y(x)+a*lambda*exp(lambda*x),y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 43.661 (sec). Leaf size: 207

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

\begin{align*} y(x)\to \frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]+2 \lambda x\right ) \left (\int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1\right )}{\exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]+\lambda x\right ) \int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1 \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]+\lambda x\right )+1} \\ y(x)\to a e^{\lambda x} \\ \end{align*}