4.9.38 \((x-y(x)) y'(x)=y(x)\)

ODE
\[ (x-y(x)) y'(x)=y(x) \] ODE Classification

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0144344 (sec), leaf count = 20

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

Maple
cpu = 0.018 (sec), leaf count = 16

\[ \left \{ x- \left (-\ln \left (y \relax (x ) \right ) +{\it \_C1} \right ) y \relax (x ) =0 \right \} \] Mathematica raw input

DSolve[(x - y[x])*y'[x] == y[x],y[x],x]

Mathematica raw output

{{y[x] -> -(x/ProductLog[-(x/E^C[1])])}}

Maple raw input

dsolve((x-y(x))*diff(y(x),x) = y(x), y(x),'implicit')

Maple raw output

x-(-ln(y(x))+_C1)*y(x) = 0