Internal problem ID [9694]
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: 29.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n} y^{2}-\lambda y+a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 78
dsolve(diff(y(x),x)=a*x^n*y(x)^2+lambda*y(x)-a*b^2*x^n*exp(2*lambda*x),y(x), singsol=all)
\[ y \relax (x ) = i \tan \left (\frac {-i \Gamma \relax (n ) b \left (-\lambda x \right )^{-n} n a \,x^{n}+i \Gamma \left (n , -\lambda x \right ) b \left (-\lambda x \right )^{-n} n a \,x^{n}+i b a \,x^{n} {\mathrm e}^{\lambda x}+c_{1} \lambda }{\lambda }\right ) b \,{\mathrm e}^{\lambda x} \]
✓ Solution by Mathematica
Time used: 1.108 (sec). Leaf size: 49
DSolve[y'[x]==a*x^n*y[x]^2+\[Lambda]*y[x]-a*b^2*x^n*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {-b^2} e^{\lambda x} \tan \left (-a \sqrt {-b^2} x^{n+1} E_{-n}(-x \lambda )+c_1\right ) \\ \end{align*}