Internal problem ID [9704]
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: 39.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n} y^{2}-\lambda y x -a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 141
dsolve(diff(y(x),x)=a*x^n*y(x)^2+lambda*x*y(x)+a*b^2*x^n*exp(lambda*x^2),y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (\frac {b a 2^{\frac {n}{2}+\frac {1}{2}} \lambda ^{-\frac {n}{2}-\frac {1}{2}} \left (-1\right )^{-\frac {n}{2}} x^{n +1} \lambda ^{\frac {n}{2}+\frac {1}{2}} \left (-1\right )^{\frac {n}{2}} \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {x^{2} \lambda }{2}\right )}{2}-\frac {b a 2^{\frac {n}{2}+\frac {1}{2}} \lambda ^{-\frac {n}{2}-\frac {1}{2}} \left (-1\right )^{-\frac {n}{2}} x^{n +1} \lambda ^{\frac {n}{2}+\frac {1}{2}} \left (-1\right )^{\frac {n}{2}} \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right )}{2}+c_{1}\right ) b \,{\mathrm e}^{\frac {x^{2} \lambda }{2}} \]
✓ Solution by Mathematica
Time used: 2.812 (sec). Leaf size: 62
DSolve[y'[x]==a*x^n*y[x]^2+\[Lambda]*x*y[x]+a*b^2*x^n*Exp[\[Lambda]*x^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {b^2} e^{\frac {\lambda x^2}{2}} \tan \left (-\frac {1}{2} a \sqrt {b^2} x^{n+1} E_{\frac {1}{2}-\frac {n}{2}}\left (-\frac {x^2 \lambda }{2}\right )+c_1\right ) \\ \end{align*}