Let

Then

Substituting these into the differential equation:

We need to match powers of x in order to write this as a single sum. Note that the last two sums already have the same power of x listed, so I will write the n = 0 term of the third summation separately, then add the last sums together:

Now for a trick. The first step is to set in the first summation. This means the sum goes from to . Since we get:

So far so good. Now for the last step of the trick: k is just a dummy variable, with no real meaning. So now let's substitute :

Now we can write the n = 0 term of the first summation then add the two sums together:

Now, if this expression is going to be true for all x, then all coefficients of xmustbe 0. Thus

for

These relations give the values of the coefficients of x when you have initial conditions. As you have not given any initial conditions to go with this equation, I can't go any further for a solution for you.

-Dan