Internal problem ID [9700]
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]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,{\mathrm e}^{\lambda x} y^{2}-k y-a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 54
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)
\[ y \relax (x ) = -\tan \left (-\Gamma \relax (k ) b a \,x^{k} \left (-\lambda x \right )^{-k}+\Gamma \left (k , -\lambda x \right ) b a \,x^{k} \left (-\lambda x \right )^{-k}+c_{1}\right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.786 (sec). Leaf size: 43
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]
\begin{align*} y(x)\to \sqrt {b^2} x^k \tan \left (-a \sqrt {b^2} x^k E_{1-k}(-x \lambda )+c_1\right ) \\ \end{align*}