Internal problem ID [9858]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-a \,{\mathrm e}^{\lambda x} f \left (x \right ) y-\lambda f \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 139
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)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{-\lambda x} c_{1} {\mathrm e}^{-\lambda x +a \left (\int f \left (x \right ) {\mathrm e}^{\lambda x}d x \right )}}{\lambda a \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int f \left (x \right ) {\mathrm e}^{\lambda x}d x \right )}}{\lambda }d x \right ) c_{1} +1\right )}-\frac {{\mathrm e}^{-\lambda x} \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int f \left (x \right ) {\mathrm e}^{\lambda x}d x \right )}}{\lambda }d x \right ) c_{1} \lambda ^{2}+\lambda ^{2}\right )}{\lambda a \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int f \left (x \right ) {\mathrm e}^{\lambda x}d x \right )}}{\lambda }d x \right ) c_{1} +1\right )} \]
✓ Solution by Mathematica
Time used: 2.065 (sec). Leaf size: 118
DSolve[y'[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 (-e^{\lambda x}-\frac {\exp \left (-\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[5])}{\lambda }\right )}{\lambda }dK[5]\right )}{\int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[6]}-\frac {a f\left (\frac {\log (K[5])}{\lambda }\right )}{\lambda }dK[5]\right )}{K[6]^2}dK[6]+c_1}\right )}{a} \\ y(x)\to -\frac {\lambda e^{\lambda (-x)}}{a} \\ \end{align*}