4.15.48 \(\left (x e^{y(x)}+e^x\right ) y'(x)+e^x y(x)+e^{y(x)}=0\)

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 = 0.0690978 (sec), leaf count = 33

\[\left \{\left \{y(x)\to c_1 e^{-x}-W\left (x e^{c_1 e^{-x}-x}\right )\right \}\right \}\]

Maple
cpu = 0.034 (sec), leaf count = 15

\[ \left \{ y \relax (x ) {{\rm e}^{x}}+x{{\rm e}^{y \relax (x ) }}+{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

y(x)*exp(x)+x*exp(y(x))+_C1 = 0