Internal problem ID [3477]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 744.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{x}+{\mathrm e}^{y} x \right ) y^{\prime }+{\mathrm e}^{x} y+{\mathrm e}^{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 31
dsolve((exp(x)+x*exp(y(x)))*diff(y(x),x)+y(x)*exp(x)+exp(y(x)) = 0,y(x), singsol=all)
\[ y \relax (x ) = -\left (\LambertW \left (x \,{\mathrm e}^{-x} {\mathrm e}^{-{\mathrm e}^{-x} c_{1}}\right ) {\mathrm e}^{x}+c_{1}\right ) {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 2.679 (sec). Leaf size: 33
DSolve[(Exp[x]+x Exp[y[x]])y'[x]+y[x] Exp[x]+Exp[y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-x}-\text {ProductLog}\left (x e^{-x+c_1 e^{-x}}\right ) \\ \end{align*}