Internal problem ID [9688]
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: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}+a b \,x^{n} {\mathrm e}^{\lambda x} y-b n \,x^{n -1}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 143
dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2-a*b*x^(n)*exp(lambda*x)*y(x)+b*n*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {c_{1} \lambda \,{\mathrm e}^{-\lambda x} {\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}}{a \left (\left (\int \lambda \,{\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right ) c_{1} +1\right )}-\frac {\left (-x^{n} \left (\int \lambda \,{\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right ) c_{1} a b -x^{n} a b \right ) {\mathrm e}^{\lambda x} {\mathrm e}^{-\lambda x}}{a \left (\left (\int \lambda \,{\mathrm e}^{\lambda x +a b \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}d x \right ) c_{1} +1\right )} \]
✓ Solution by Mathematica
Time used: 61.176 (sec). Leaf size: 109
DSolve[y'[x]==a*Exp[\[Lambda]*x]*y[x]^2-a*b*x^(n)*Exp[\[Lambda]*x]*y[x]+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to b \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n-\frac {c_1 \lambda \left (e^{\lambda x}\right )^{-\frac {a b \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n \operatorname {ExpIntegralE}\left (-n,-\log \left (e^{x \lambda }\right )\right )}{\lambda }}}{a+a c_1 \int _1^{e^{x \lambda }}K[1]^{-\frac {a b \operatorname {ExpIntegralE}(-n,-\log (K[1])) \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }}dK[1]} \\ \end{align*}