Internal problem ID [9690]
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: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-{\mathrm e}^{\lambda x} y^{2}-a \,x^{n} y-a \lambda \,x^{n} {\mathrm e}^{-\lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 89
dsolve(diff(y(x),x)=exp(lambda*x)*y(x)^2+a*x^(n)*y(x)+a*lambda*x^n*exp(-lambda*x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int {\mathrm e}^{\frac {x \left (a \,x^{n}-\lambda n -\lambda \right )}{n +1}}d x \right ) \lambda +c_{1} \lambda +{\mathrm e}^{\frac {x \left (a \,x^{n}-\lambda n -\lambda \right )}{n +1}}\right ) {\mathrm e}^{-\lambda x}}{c_{1}+\int {\mathrm e}^{\frac {x \left (a \,x^{n}-\lambda n -\lambda \right )}{n +1}}d x} \]
✓ Solution by Mathematica
Time used: 1.735 (sec). Leaf size: 254
DSolve[y'[x]==Exp[\[Lambda]*x]*y[x]^2+a*x^(n)*y[x]+a*\[Lambda]*x^n*Exp[-\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {a x^{n+1}}{n+1}}}{\left (\lambda +e^{x \lambda } K[2]\right )^2}-\int _1^x\left (\frac {2 e^{\frac {a K[1]^{n+1}}{n+1}} \left (a \lambda K[1]^n+a e^{\lambda K[1]} K[2] K[1]^n+e^{2 \lambda K[1]} K[2]^2\right )}{\left (\lambda +e^{\lambda K[1]} K[2]\right )^3}-\frac {e^{\frac {a K[1]^{n+1}}{n+1}-\lambda K[1]} \left (a e^{\lambda K[1]} K[1]^n+2 e^{2 \lambda K[1]} K[2]\right )}{\left (\lambda +e^{\lambda K[1]} K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {e^{\frac {a K[1]^{n+1}}{n+1}-\lambda K[1]} \left (a \lambda K[1]^n+a e^{\lambda K[1]} y(x) K[1]^n+e^{2 \lambda K[1]} y(x)^2\right )}{\left (\lambda +e^{\lambda K[1]} y(x)\right )^2}dK[1]=c_1,y(x)\right ] \]