61.19.12 problem 12

Internal problem ID [12281]
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 : 12
Date solved : Tuesday, January 28, 2025 at 02:08:51 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 92

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

\[ y = \frac {-c_{1} {\mathrm e}^{-2 \lambda x +a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )}-\left (\int {\mathrm e}^{-\lambda x +a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )}d x \right ) {\mathrm e}^{-\lambda x} c_{1} \lambda -\lambda ^{2} {\mathrm e}^{-\lambda x}}{a \left (\left (\int {\mathrm e}^{-\lambda x +a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )}d x \right ) c_{1} +\lambda \right )} \]

✓ Solution by Mathematica

Time used: 2.415 (sec). Leaf size: 166

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

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