Using Frobenius theorem

[shemer77]

Homework Statement

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

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)c₀x^r-1+$\sum$(C_n+1(n+1+r)(n+r) + C_n+1(n+1+r)+C_n)x^n+r

now if you solve for r you get
r^2-r+r and r_1,2=0

you put this into this
C_n+1(n+1+r)(n+r) + C_n+1(n+1+r)+C_n=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 don't think I did because I double checked my work)?

 

[mighty2000]

Thank you for sharing your attempt at solving this problem. It seems like you have made good progress in using the method of Frobenius to obtain a series solution for the given differential equation. However, it is important to note that the indicial roots of a regular singular point do not necessarily have to differ by an integer. It is possible for them to differ by a non-integer value, in which case the method of Frobenius may not work. Can you provide more information or context about the specific problem you are working on? It would be helpful to see the full differential equation and the steps you took to obtain your solution.

In terms of your question about the recursion formula, it is possible that you made a mistake in your calculations. I would suggest double checking your work and also trying to solve the recursion formula for y2 as well to see if you encounter the same issue. If you are still having trouble, it would be helpful to see the full problem and your work in order to provide more specific guidance.