Internal problem ID [3167]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 15
Problem number: 421.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y y^{\prime }+x \,{\mathrm e}^{-x} \left (1+y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 46
dsolve(y(x)*diff(y(x),x)+x*exp(-x)*(1+y(x)) = 0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\left (\LambertW \left (-{\mathrm e}^{c_{1}-1-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}}\right ) {\mathrm e}^{x}-c_{1} {\mathrm e}^{x}+{\mathrm e}^{x}+x +1\right ) {\mathrm e}^{-x}}-1 \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 32
DSolve[y[x] y'[x]+x Exp[-x](1+y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1-\text {ProductLog}\left (-e^{-e^{-x} \left (x+(1+c_1) e^x+1\right )}\right ) \\ \end{align*}