ODE
\[ y(x) y'(x)+e^{-x} x (y(x)+1)=0 \] ODE Classification
[_separable]
Book solution method
Linear ODE
Mathematica ✓
cpu = 0.0358439 (sec), leaf count = 32
\[\left \{\left \{y(x)\to -W\left (-e^{-e^{-x} \left (\left (c_1+1\right ) e^x+x+1\right )}\right )-1\right \}\right \}\]
Maple ✓
cpu = 0.009 (sec), leaf count = 24
\[ \left \{ -\ln \left ( 1+y \left ( x \right ) \right ) + \left ( -x-1 \right ) {{\rm e}^{-x}}+{\it \_C1}+y \left ( x \right ) =0 \right \} \] 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),'implicit')
Maple raw output
-ln(1+y(x))+(-x-1)*exp(-x)+_C1+y(x) = 0