Using Frobenius theorem

1. Jul 23, 2012

shemer77

1. The problem statement, all variables and given/known data
In Problems 25–30, x=0 is a regular singular point of
the given differential equation. Show that the indicial
roots of the singularity differ by an integer. Use the method
of Frobenius to obtain at least one series solution about
x=0.
http://gyazo.com/ef4d819c3a6f0b820f1dfc16e01889d2

3. The attempt at a solution
If someone wants me to I will right out what I did but basically what it came out to is
(r(r-1)+r)c0xr-1+$\sum$(Cn+1(n+1+r)(n+r) + Cn+1(n+1+r)+Cn)xn+r

now if you solve for r you get
r^2-r+r and r1,2=0

you put this into this
Cn+1(n+1+r)(n+r) + Cn+1(n+1+r)+Cn=0 and rearrange it a bit to make it simple and you got the setup for a recursion formula

Now if you try using the recursion formula to solve for a y1 you will see that when you let n=0 you will get a 0 on the bottom making it undefined
and I know you can use this
http://gyazo.com/a3e25290c490344eae98c9b7f9e59f3f
if this happens in y2 but what happens when your trying to solve for y1 and this happens?

OR did I mess up somewhere(I dont think I did because I double checked my work)?