Solve the equation y' (y+1) = xy - y dif from the separable equation

1. Rewrite the given equation y' (y+1) = xy - y as y' = \frac{y (x - 1)}{y+1} .
2. Separate variables \frac{y+1}{y} \, dy = (x - 1) \, dx .
3. Integrate both sides \int \frac{y+1}{y} \, dy = \int (x - 1) \, dx .
4. Split and integrate the left-hand side \int 1 \, dy + \int \frac{1}{y} \, dy = \int x \, dx - \int 1 \, dx .
5. Solve the integrals y + \ln|y| = \frac{x^2}{2} - x + C .

Answer: y + \ln|y| = \frac{x^2}{2} - x + C