Internal problem ID [10451]
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]
\[ \boxed {y^{\prime }-a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}-\left ({\mathrm e}^{\lambda x} x^{n} b -\lambda \right ) y=c \,x^{n}} \]
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left (x^{n} \left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right ) b n -x^{n} \Gamma \left (n \right ) \left (-\lambda x \right )^{-n} b n +x^{n} {\mathrm e}^{\lambda x} b +c_{1} \lambda \right )}{2 b^{2} \lambda }\right ) \sqrt {4 b^{2} a c -b^{4}}-b^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a b} \]
✓ Solution by Mathematica
Time used: 3.112 (sec). Leaf size: 102
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]
\[ \text {Solve}\left [\int _1^{\sqrt {\frac {a e^{2 x \lambda }}{c}} y(x)}\frac {1}{K[1]^2-\sqrt {\frac {b^2}{a c}} K[1]+1}dK[1]=\frac {c x^n e^{\lambda (-x)} (\lambda (-x))^{-n} \sqrt {\frac {a e^{2 \lambda x}}{c}} \Gamma (n+1,-x \lambda )}{\lambda }+c_1,y(x)\right ] \]