4.9.44 \((x-y(x)) y'(x)=\left (e^{-\frac {x}{y(x)}}+1\right ) y(x)\)

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

\[ \left \{ -{\it \_C1}+\ln \left ({\frac {1}{x} \left (y \relax (x ) {{\rm e}^{{\frac {x}{y \relax (x ) }}}}+x \right ) } \right ) +\ln \relax (x ) =0 \right \} \] 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