Internal problem ID [10444]
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]
\[ \boxed {y^{\prime } x -a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}-\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y={\mathrm e}^{\lambda x} c} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 86
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)
\[ y \left (x \right ) = -\frac {\left (\tan \left (\frac {\left (x^{n} b \left (\Gamma \left (n , -x \lambda \right )-\Gamma \left (n \right )\right ) \left (-x \lambda \right )^{-n}-c_{1} \right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2}}\right ) \sqrt {4 a \,b^{2} c -b^{4}}+b^{2}\right ) x^{-n}}{2 a b} \]
✓ Solution by Mathematica
Time used: 3.62 (sec). Leaf size: 87
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]
\[ \text {Solve}\left [\int _1^{\sqrt {\frac {a x^{2 n}}{c}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {b^2}{a c}} K[1]+1}dK[1]=-c (\lambda (-x))^{-n} \sqrt {\frac {a x^{2 n}}{c}} \Gamma (n,-x \lambda )+c_1,y(x)\right ] \]