Solve (x-1)y''-xy'+y=0 , given x>1 and y1=ex
The Attempt at a Solution
I've tried solving it by multiplying everythin for (1/x-1), so my equation looks like:
y''-(x/x-1)y'+y=0 (because x-1 is unequal to 0) (1)
so now the equation has the form of:
y''+p(x)y'+y=0, with p(x)=(-x/x-1).
Then, given the fact that y1 is a solution, then another solution has the form Y(x)=v(x)y1
Now, after replacing the y factors in (1) with Y and its derivatives, I get:
Which can me solved as a first order differential equation by doing v''(x)=z'(x) and v'(x)=z(x).
Then, let μ(x)=e∫p(x)dx=e∫(-x/x-1)dx=e-x-ln(x-1)=e-x(1/x-1)
Now, because z(x)=v'(x)→∫z(x)dx=v(x)
Then I have v(x)=c1∫(ex(x-1)dx=c1((x-2)ex+c2)
But, whenever I do Y(x)=v(x)y1, and replace it in (1), it doesn't satisfy the equation-
Can somebody help?