Internal problem ID [10447]
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]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-y \lambda x=a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 90
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)
\[ y \left (x \right ) = -\tan \left (-a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right ) \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}}+a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (-x^{2} \lambda \right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {x^{2} \lambda }{2}\right )+c_{1} \right ) b \,{\mathrm e}^{\frac {x^{2} \lambda }{2}} \]
✓ Solution by Mathematica
Time used: 2.366 (sec). Leaf size: 83
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]
\[ y(x)\to \sqrt {b^2} e^{\frac {\lambda x^2}{2}} \tan \left (a \sqrt {b^2} \lambda 2^{\frac {n-1}{2}} x^{n+3} \left (\lambda \left (-x^2\right )\right )^{-\frac {n}{2}-\frac {3}{2}} \Gamma \left (\frac {n+1}{2},-\frac {x^2 \lambda }{2}\right )+c_1\right ) \]