ODE
\[ \left (x e^{y(x)}+e^x\right ) y'(x)+e^x y(x)+e^{y(x)}=0 \] ODE Classification
[_exact]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.409458 (sec), leaf count = 33
\[\left \{\left \{y(x)\to c_1 e^{-x}-W\left (x e^{-x+c_1 e^{-x}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.077 (sec), leaf count = 31
\[[y \left (x \right ) = -\left (\LambertW \left (x \,{\mathrm e}^{-x} {\mathrm e}^{-\textit {\_C1} \,{\mathrm e}^{-x}}\right ) {\mathrm e}^{x}+\textit {\_C1} \right ) {\mathrm e}^{-x}]\] Mathematica raw input
DSolve[E^y[x] + E^x*y[x] + (E^x + E^y[x]*x)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]/E^x - ProductLog[E^(-x + C[1]/E^x)*x]}}
Maple raw input
dsolve((exp(x)+x*exp(y(x)))*diff(y(x),x)+y(x)*exp(x)+exp(y(x)) = 0, y(x))
Maple raw output
[y(x) = -(LambertW(x/exp(x)/exp(1/exp(x)*_C1))*exp(x)+_C1)/exp(x)]