Using the Frobenius Method - 2nd order DE

[outhsakotad]

Using the Frobenius Method -- 2nd order DE

Homework Statement

y"+(1/sinx)y'+((1-x)/x^2)y=0

Find the indicial equation and forms of two linearly independent expansions about x=0. Don't find the coefficents.

Homework Equations

The Attempt at a Solution

The singular points at n*pi are regular, and thus I need to use the method of Frobenius. I normally don't have a problem using this method, but I am a bit thrown off by the (1/sinx) term. I tried rewriting the DE as (x^2)*sin(x)*y"+(x^2)*y'+sin(x)*(1-x)y=0, thinking that it might be a good idea to write sin(x) as an expansion. But then I end up with a double summation, and I'm not sure how to get the indicial equation out of it first of all. And I am not sure what it means by "find the forms of the two linearly independent expansions..." Anyway, I attached a PDF showing my work and where I got stuck. Any hints would be greatly appreciated.

 

[outhsakotad]

Never mind.