Internal problem ID [10443]
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: 35.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x -a \,{\mathrm e}^{\lambda x} y^{2}-k y=a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 38
dsolve(x*diff(y(x),x)=a*exp(lambda*x)*y(x)^2+k*y(x)+a*b^2*x^(2*k)*exp(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = -\tan \left (a b \,x^{k} \left (\Gamma \left (k , -x \lambda \right )-\Gamma \left (k \right )\right ) \left (-x \lambda \right )^{-k}+c_{1} \right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.593 (sec). Leaf size: 47
DSolve[x*y'[x]==a*Exp[\[Lambda]*x]*y[x]^2+k*y[x]+a*b^2*x^(2*k)*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {b^2} x^k \tan \left (-a \sqrt {b^2} x^k (\lambda (-x))^{-k} \Gamma (k,-x \lambda )+c_1\right ) \]