Internal problem ID [7921]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 341.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {\left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+y \,{\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 31
dsolve((x*exp(y(x))+exp(x))*diff(y(x),x)+exp(y(x))+y(x)*exp(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: 4.424 (sec). Leaf size: 33
DSolve[E^y[x] + E^x*y[x] + (E^x + E^y[x]*x)*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*}