61.4.17 problem 38

Internal problem ID [12122]
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 : 38
Date solved : Tuesday, January 28, 2025 at 12:25:51 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 45

dsolve(diff(y(x),x)=a*exp(-lambda*x^2)*y(x)^2+lambda*x*y(x)+a*b^2,y(x), singsol=all)
 

\[ y = \tan \left (\frac {a b \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{2}\right )-2 c_{1} \sqrt {\lambda }}{2 \sqrt {\lambda }}\right ) b \,{\mathrm e}^{\frac {\lambda \,x^{2}}{2}} \]

✓ Solution by Mathematica

Time used: 1.401 (sec). Leaf size: 63

DSolve[D[y[x],x]==a*Exp[-\[Lambda]*x^2]*y[x]^2+\[Lambda]*x*y[x]+a*b^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {b^2} e^{\frac {\lambda x^2}{2}} \tan \left (\frac {\sqrt {\frac {\pi }{2}} a \sqrt {b^2} \text {erf}\left (\frac {\sqrt {\lambda } x}{\sqrt {2}}\right )}{\sqrt {\lambda }}+c_1\right ) \]