Let

Then

and

So your differential equation reads:

We need to match the powers of x in each summation. So in the first summation, let

where in this last step I have replaced the dummy variable k with an n. (Yes, this is legitimate.)

So the differential equation now reads:

Now we need all the summations to start with the same number. So write out the extra terms:

Now combine all the summations into one:

Now match the coefficients of powers of x on both sides of the equation. I.e. all the coefficients of powers of x on the right hand side are 0, so

and

You can work it out from there. (And please check to make sure I didn't make a typo or some other error in there. The details for this kind of problem are killers!)

-Dan