Internal problem ID [9701]
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: 36.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}-\left ({\mathrm e}^{\lambda x} x^{n} b -n \right ) y-{\mathrm e}^{\lambda x} c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 97
dsolve(x*diff(y(x),x)=a*x^(2*n)*exp(lambda*x)*y(x)^2+(b*x^n*exp(lambda*x)-n)*y(x)+c*exp(lambda*x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (\Gamma \left (n , -\lambda x \right ) b \,x^{n} \left (-\lambda x \right )^{-n}-\Gamma \relax (n ) b \,x^{n} \left (-\lambda x \right )^{-n}-c_{1}\right )}{2 b^{2}}\right ) \sqrt {4 b^{2} a c -b^{4}}+b^{2}\right ) x^{-n}}{2 a b} \]
✓ Solution by Mathematica
Time used: 2.909 (sec). Leaf size: 133
DSolve[x*y'[x]==a*x^(2*n)*Exp[\[Lambda]*x]*y[x]^2+(b*x^n*Exp[\[Lambda]*x]-n)*y[x]+c*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {c} \sqrt {\frac {b^2}{a c}}+\sqrt {4 a c-b^2} \tan \left (\frac {\sqrt {4 a c-b^2} \left (-c \sqrt {\frac {a x^{2 n}}{c}} E_{1-n}(-x \lambda )+c_1\right )}{2 \sqrt {a} \sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {\frac {a x^{2 n}}{c}}} \\ \end{align*}