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I'm having trouble with this one.

## Homework Statement

Find a particular solution of the second-order homogeneous lineal differential equation

[itex] x^2y'' + xy' - y = 0[/itex]

taking in account that [itex] x = 0 [/itex] is a regular singular point

*and performing a power series expansion*.

## Homework Equations

[itex] x^2y'' + xy' - y = 0[/itex]

## The Attempt at a Solution

I see that the equation given is an Euler equation, but the question asks for a power series solution, so i tried with the Frobenius method. Assuming there is at least one solution with the form [itex] y = x^\sigma\sum{a_nx^n} [/itex].

First, I divide the whole differential equation by [itex] x^2[/itex]. Then I substitute the expression above so I get

[itex]\sum{(n+\sigma)(n+\sigma-1)a_nx^{n+\sigma-2}} + \frac{1}{x}\sum{(n+\sigma)a_nx^{n+\sigma-1}}-\frac{1}{x^2}\sum{a_nx^{n+\sigma}} [/itex]

And now, dividing by [itex] x^{\sigma-2} [/itex], I get

[itex] \sum{((n+\sigma)(n+\sigma-1)+(n+\sigma) - 1})a_nx^n [/itex]

Now I don't know how to find the recurrence relation I'm looking for in order to find the form of [itex] a_n [/itex]. In all the examples I've been able to find, in the last expression one always finds terms of [itex]a_{n-1}[/itex], for example, but here I don't know how to continue.

Did I do something wrong? I tried to follow the steps given in my textbook. I'm confused because I believe the equation given fits the requeriments needed in order to the Frobenius method to be applicable, but this happens to me every time I try to solve an Euler equation using it.

Thank you very much in advance.