Power Series to solve Second order Differential Equations

[xWaffle]

Homework Statement

When solving a D.E. with power series, I've encountered something along the lines of:

(2 - r)^{2}g'' = -2

Homework Equations

Power Series

The Attempt at a Solution

I know I am just supposed to assume such a series exists, and work from there. But I'm really getting fudged up when it comes to factoring the r's back into the series, which makes the r's in the summation 2 powers higher than just 'n'.

I've reached the end where I have a recursive definition of a_n+1 in terms of a_n, but there are ridiculous fractions on the coefficients that I don't think I can generalise for values of n. They don't seem to have a pattern, and that's..a problem.

Is there an easier way to solve this, leaving the r-polynomial as it is? I expanded it and applied g'' to each term, then subbed in the second derivative of the series that represents g(x) [the standard assumed series, a_nrⁿ]

 

[haruspex]

This is just g = g(r), right?
Do you have to use power series? Looks more straightforward to get it into the form g'' = ... and integrate twice.
If you wish to persevere with power series, please post your working.

 

[xWaffle]

It's an assignment, I'm much more than capable of solving this with more traditional means, but I can't submit that work.

I don't have my notebook at the moment but from memory, when using power series, you assume

y = \sum_{n=0}^{\infty} a_{n}x^{n}

Which means

y' = \sum_{n=1}^{\infty} na_{n}x^{n-1}
y'' = \sum_{n=2}^{\infty} n(n-1)a_{n}x^{n-2}

And particularly for the second derivative, re-write the sum so the power of x is the same as for y

y'' = \sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}x^{n}

Now, after putting the sum forms of y and its derivatives back into the equation, I think I fiddled around with it by expanding whatever quantity was multiplied with y'', and replaced all occurrences of y and its derivatives with the appropriate sums.

I guess you could re-write the r's as x's since I wrote x's in the sums instead of r's, so you'd need to factor that variable into the sums when appropriate, which messes with the powers of x again, even though I just fixed them before.

This also leads to problematic situations where different sums are -beginning- at different indices of n. I know I'm supposed to pop out the first few terms til the sum starts at the same index as the others, but I'm unsure about how to do this procedure.

I know the end goal is to get all sums starting at the same index, and all x's in the sums to have the same power, so I can factor out the x's and have one large sum of coefficients.

My problem is, I don't exactly know how to shift indices correctly (because of the a's within the sums) and I don't really know how to take out the first term or two to get the starting index to match with the other sums. I also don't know what I'm supposed to actually do with those coefficients once I get them out of their mother sums.