Method of Frobenius for Solving ODE at x=0 | Step-by-Step Guide

[bart007]

Homework Statement

Using method of frobenius about x=0 to solve:
(1-x) y''+xy'-\frac{\alpha^2}{x^2}+=0

Homework Equations

N/A

The Attempt at a Solution

1. plug in series into the equation.
2. adjust the index off all the terms.
3. write the extra terms separately so that we have all series starting at the same point.

and I get...

that the roots from the indicial equation
\lambda=\frac{1}{2}(1\pm\sqrt{4a^2+1})
and for the recursion I get
c_n=\frac{c_{n-1}(\lambda+n-1)(\lambda+n-2)-c_{n-2}(\lambda+n-2)}{(\lambda+k)(\lambda+n-1)-\alpha^2}

I am not sure how to check if this is correct or not.
Using a number for \alpha I don't get any nice series that has a function.

 

[bart007]

Sorry for the double post, first post was not showing.