Internal problem ID [9699]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}-y \lambda x -a \,b^{2}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 49
dsolve(diff(y(x),x)=a*exp(-lambda*x^2)*y(x)^2+lambda*x*y(x)+a*b^2,y(x), singsol=all)
\[ y \left (x \right ) = -\tan \left (\frac {-b a \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{2}\right )+2 c_{1} \sqrt {\lambda }}{2 \sqrt {\lambda }}\right ) b \,{\mathrm e}^{\frac {x^{2} \lambda }{2}} \]
✓ Solution by Mathematica
Time used: 1.46 (sec). Leaf size: 63
DSolve[y'[x]==a*Exp[-\[Lambda]*x^2]*y[x]^2+\[Lambda]*x*y[x]+a*b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}