61.4.9 problem 30

Internal problem ID [12114]
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
Date solved : Tuesday, January 28, 2025 at 12:25:08 AM
CAS classification : [_Riccati]

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

✗ 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: 46.023 (sec). Leaf size: 217

DSolve[D[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],x,IncludeSingularSolutions -> True]
 

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