4.9.38 $(x-y(x))y^{\prime }(x)=y(x)$

ODE
$$(x-y(x))y^{\prime }(x)=y(x)$$ ODE Classification

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

Book solution method
Homogeneous equation

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

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

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

$$\{x-(-\ln \left(y\left(x\right)\right)+\mathit{\_}\mathit{C}\mathit{1})y\left(x\right)=0\}$$ 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