ODE
\[ y(x) y'(x)+e^{-x} x (y(x)+1)=0 \] ODE Classification
[_separable]
Book solution method
Linear ODE
Mathematica ✓
cpu = 0.206088 (sec), leaf count = 32
\[\left \{\left \{y(x)\to -1-W\left (-e^{-e^{-x} \left (x+(1+c_1) e^x+1\right )}\right )\right \}\right \}\]
Maple ✓
cpu = 0.187 (sec), leaf count = 46
\[[y \left (x \right ) = {\mathrm e}^{-\left (\LambertW \left (-{\mathrm e}^{\textit {\_C1} -1-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}}\right ) {\mathrm e}^{x}-\textit {\_C1} \,{\mathrm e}^{x}+{\mathrm e}^{x}+x +1\right ) {\mathrm e}^{-x}}-1]\] Mathematica raw input
DSolve[(x*(1 + y[x]))/E^x + y[x]*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -1 - ProductLog[-E^(-((1 + x + E^x*(1 + C[1]))/E^x))]}}
Maple raw input
dsolve(y(x)*diff(y(x),x)+x*exp(-x)*(1+y(x)) = 0, y(x))
Maple raw output
[y(x) = exp(-(LambertW(-exp(_C1-1-x*exp(-x)-exp(-x)))*exp(x)-_C1*exp(x)+exp(x)+x+1)*exp(-x))-1]