Internal problem ID [9687]
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]
\[ \boxed {y^{\prime }+\lambda \,{\mathrm e}^{\lambda x} y^{2}-a \,x^{n} {\mathrm e}^{\lambda x} y+x^{n} a=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 135
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)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\lambda x} {\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}}{\lambda ^{2} \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}}{\lambda }d x \right ) c_{1} +1\right )}+\frac {{\mathrm e}^{-\lambda x} \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}}{\lambda }d x \right ) c_{1} \lambda ^{2}+\lambda ^{2}\right )}{\lambda ^{2} \left (\left (\int \frac {{\mathrm e}^{-\lambda x +a \left (\int {\mathrm e}^{\lambda x} x^{n}d x \right )}}{\lambda }d x \right ) c_{1} +1\right )} \]
✓ Solution by Mathematica
Time used: 3.295 (sec). Leaf size: 110
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 e^{-2 \lambda x} \left (e^{\lambda x}+\frac {\left (e^{\lambda x}\right )^{-\frac {a \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n \operatorname {ExpIntegralE}\left (-n,-\log \left (e^{x \lambda }\right )\right )}{\lambda }}}{\int _1^{e^{x \lambda }}K[1]^{-\frac {a \operatorname {ExpIntegralE}(-n,-\log (K[1])) \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }-2}dK[1]+c_1}\right ) \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}