61.4.18 problem 39

Internal problem ID [12123]
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
Date solved : Tuesday, January 28, 2025 at 12:25:53 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.033 (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)
 

\[ y = -\tan \left (-a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \Gamma \left (\frac {n}{2}+\frac {1}{2}\right ) \left (-\lambda \,x^{2}\right )^{-\frac {n}{2}-\frac {1}{2}}+a b 2^{\frac {n}{2}-\frac {1}{2}} x^{n +1} \left (-\lambda \,x^{2}\right )^{-\frac {n}{2}-\frac {1}{2}} \Gamma \left (\frac {n}{2}+\frac {1}{2}, -\frac {\lambda \,x^{2}}{2}\right )+c_{1} \right ) b \,{\mathrm e}^{\frac {\lambda \,x^{2}}{2}} \]

✓ Solution by Mathematica

Time used: 1.468 (sec). Leaf size: 83

DSolve[D[y[x],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]
 

\[ y(x)\to \sqrt {b^2} e^{\frac {\lambda x^2}{2}} \tan \left (a \sqrt {b^2} \lambda 2^{\frac {n-1}{2}} x^{n+3} \left (\lambda \left (-x^2\right )\right )^{-\frac {n}{2}-\frac {3}{2}} \Gamma \left (\frac {n+1}{2},-\frac {x^2 \lambda }{2}\right )+c_1\right ) \]