Differential Equation – Solving $xy(x)y'(x) = y(x) – 1$

Question

I am given the differential equation:
$$ xyy' = y – 1 $$
I have tried to solve it by separating the variables, but I have only come this far:
$$ \int 1 + \frac{1}{y-1} dy = \int \frac{1}{x} dx = y + \log|y-1| = \log{|x|} + C $$
I am struggeling on how to isolate $ y(x) $. Can somebody please give me a hint on how to continue?

Answer 1

The lambert W function is perfectly valid in this case: $$ y+\ln(y-1)=\ln x+C\implies e^y(y-1)=xC_1\implies y-1=W_n\left(\frac{xC_1}{e}\right)\\\implies y(x) = W_n\left({xC_2}\right)+1 $$