Reduction of Order not giving a second solution

[1MileCrash]

Homework Statement

(x-1)y'' - xy' + y = 0, y=e^x is a solution

Homework Equations

The Attempt at a Solution

Assume the second solution is of the form ve^x, where v' = (y^'2)e^-int[-x/(x-1)]

So v' = e^(-2x)e^(x+ln|x-1|) = e^(ln|x+1|-x)

Then, this second solution must be

(e^x)(e^(ln|x-1|-x))

=e^(ln|x-1)

=x-1

But, this is no solution to the DE. What went wrong?

Thank you.
NEVERMIND, I forgot to integrate v'. What was I thinking?

 

[Simon Bridge]

You mean: $$y^{\prime\prime} - \frac{x}{x-1}y^\prime + \frac{1}{x-1}y = 0$$ $$\frac{d}{dx}\left ( v^\prime(x)e^{2x}e^{\int \frac{xdx}{x-1}} \right )=0$$