Internal problem ID [10434]
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: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+{\mathrm e}^{\lambda x} y^{2} \lambda -a \,x^{n} y \,{\mathrm e}^{\lambda x}=-x^{n} a} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 92
dsolve(diff(y(x),x)=-lambda*exp(lambda*x)*y(x)^2+a*x^(n)*exp(lambda*x)*y(x)-a*x^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-x \lambda } \left (\int {\mathrm e}^{-x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} \lambda +\lambda ^{2} {\mathrm e}^{-x \lambda }+c_{1} {\mathrm e}^{-2 x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}}{\lambda \left (\left (\int {\mathrm e}^{-x \lambda +a \left (\int {\mathrm e}^{x \lambda } x^{n}d x \right )}d x \right ) c_{1} +\lambda \right )} \]
✓ Solution by Mathematica
Time used: 6.627 (sec). Leaf size: 185
DSolve[y'[x]==-\[Lambda]*Exp[\[Lambda]*x]*y[x]^2+a*x^(n)*Exp[\[Lambda]*x]*y[x]-a*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-2 \lambda x} \left (e^{\lambda x} \int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a \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 \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}\right )}{\int _1^{e^{x \lambda }}\frac {\exp \left (\frac {a \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} \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}