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## Homework Statement

This is for a differential equations class I'm taking and we're talking about the method of Frobeneus, Euler equations, and power series solutions for non-constant coefficients. The ODE is:

[tex]6x^2y''+7xy'-(1-x^2)y=0[/tex]

I need to find the recurrence formula and I keep running into a problem with my grouping. So the question I have is when I take the derivative of a sum, does that change its index, or does the index stay the same? Because when the book explains it with power series the index is changed with each derivative taken. But when they do the Frobeneus method the index does not change when a derivative is taken.

## Homework Equations

## The Attempt at a Solution

Assume $$y=\sum_{n=0}^{\infty}a_nx^{n+r}$$ where r is the root of the indicial and then the derivatives are:

$$y'=\sum_{n=1}^{\infty}(n+r)a_nx^{n+r-1}$$ and $$y''=\sum_{n=2}^{\infty}(n+r-1)(n+r)a_nx^{n+r-2}$$

Plugging this into the ODE gives:

$$6\sum_{n=2}^{\infty}(n+r-1)(n+r)a_nx^{n+r} +7\sum_{n=1}^{\infty}(n+r)a_nx^{n+r}-\sum_{n=0}^{\infty}a_nx^{n+r}+\sum_{n=0}^{\infty}a_nx^{n+r+2}$$

Here is where the problem begins, I've tried changing my indices, but I can't quite get the powers of x and the indices to agree, and when I do get the indices to agree I get three different values of a (##a_1##,##a_2##, and ##a_0##)