61.4.5 problem 26

Internal problem ID [12110]
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
Date solved : Tuesday, January 28, 2025 at 12:24:51 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n} \end{align*}

✓ Solution by Maple

Time used: 0.002 (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)
 

\[ y = \frac {{\mathrm e}^{-\lambda x} \left (\int {\mathrm e}^{-\lambda x +a \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}d x \right ) c_{1} \lambda +\lambda ^{2} {\mathrm e}^{-\lambda x}+c_{1} {\mathrm e}^{-2 \lambda x +a \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}}{\lambda \left (\left (\int {\mathrm e}^{-\lambda x +a \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}d x \right ) c_{1} +\lambda \right )} \]

✓ Solution by Mathematica

Time used: 3.340 (sec). Leaf size: 158

DSolve[D[y[x],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 (\exp \left (-\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )+e^{\lambda x} \int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1 e^{\lambda x}\right )}{\int _1^{e^{x \lambda }}\frac {\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )}{K[2]^2}dK[2]+c_1} \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}