Internal problem ID [10438]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and
exponential functions
Problem number: 30.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y=b \lambda \,{\mathrm e}^{\lambda x}} \]
✗ Solution by Maple
dsolve(diff(y(x),x)=a*x^n*y(x)^2-a*b*x^n*exp(lambda*x)*y(x)+b*lambda*exp(lambda*x),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 53.05 (sec). Leaf size: 190
DSolve[y'[x]==a*x^n*y[x]^2-a*b*x^n*Exp[\[Lambda]*x]*y[x]+b*\[Lambda]*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b e^{2 \lambda x} \left (\int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a b \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }\right )}{K[1]^2}dK[1]+c_1\right )}{e^{\lambda x} \int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a b \Gamma (n+1,-\log (K[1])) (-\log (K[1]))^{-n} \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }\right )}{K[1]^2}dK[1]+\exp \left (\frac {a b \left (-\log \left (e^{\lambda x}\right )\right )^{-n} \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n \Gamma \left (n+1,-\log \left (e^{x \lambda }\right )\right )}{\lambda }\right )+c_1 e^{\lambda x}} \\ y(x)\to b e^{\lambda x} \\ \end{align*}