Hello everyone!

Here's the problem. Equation

needs to be solved using Frobenius method at

.

First, I've translated the origin,

, so that the original equation becomes

,

derivatives staying the same,

Now, we seek solution in the form

Upon differentiating and substituting, we get

I hope I've done everything right by now. Now, I'm not sure if my indicial equation is right. First, I shifted the second sum so it becomes

I've taken

to be indicial eqation and then plugged the solutions (

and

). Recurrence formula for

gives trivial solution

and formula for

gives polynomial solution.

But, if I use

as a solution, I get

in the polynomial and it diverges for

. However, if I start the sum with index 1, the solution seems right.

So, I'm just wondering if I did everyting right, especially with indicial equation.

Thanks in advance!