ODE
\[ (x-y(x)) y'(x)=\left (e^{-\frac {x}{y(x)}}+1\right ) y(x) \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.0376383 (sec), leaf count = 23
\[\left \{\left \{y(x)\to -\frac {x}{W\left (\frac {x}{x-e^{c_1}}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.033 (sec), leaf count = 26
\[ \left \{ -{\it \_C1}+\ln \left ( {\frac {1}{x} \left ( y \left ( x \right ) {{\rm e}^{{\frac {x}{y \left ( x \right ) }}}}+x \right ) } \right ) +\ln \left ( x \right ) =0 \right \} \] Mathematica raw input
DSolve[(x - y[x])*y'[x] == (1 + E^(-(x/y[x])))*y[x],y[x],x]
Mathematica raw output
{{y[x] -> -(x/ProductLog[x/(-E^C[1] + x)])}}
Maple raw input
dsolve((x-y(x))*diff(y(x),x) = (exp(-x/y(x))+1)*y(x), y(x),'implicit')
Maple raw output
-_C1+ln((y(x)*exp(x/y(x))+x)/x)+ln(x) = 0