Internal problem ID [9698]
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: 33.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}-\left ({\mathrm e}^{\lambda x} x^{n} b -\lambda \right ) y-c \,x^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 114
dsolve(diff(y(x),x)=a*x^n*exp(2*lambda*x)*y(x)^2+(b*x^n*exp(lambda*x)-lambda)*y(x)+c*x^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (\Gamma \left (n , -\lambda x \right ) b n \,x^{n} \left (-\lambda x \right )^{-n}-\Gamma \relax (n ) b n \,x^{n} \left (-\lambda x \right )^{-n}+x^{n} {\mathrm e}^{\lambda x} b +c_{1} \lambda \right )}{2 b^{2} \lambda }\right ) \sqrt {4 b^{2} a c -b^{4}}-b^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a b} \]
✓ Solution by Mathematica
Time used: 2.642 (sec). Leaf size: 150
DSolve[y'[x]==a*x^n*Exp[2*\[Lambda]*x]*y[x]^2+(b*x^n*Exp[\[Lambda]*x]-\[Lambda])*y[x]+c*x^n,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 {e^{\lambda (-x)} \sqrt {4 a c-b^2} \left (-c x^{n+1} \sqrt {\frac {a e^{2 \lambda x}}{c}} E_{-n}(-x \lambda )+c_1 e^{\lambda x}\right )}{2 \sqrt {a} \sqrt {c}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {\frac {a e^{2 \lambda x}}{c}}} \\ \end{align*}