Consider the differential equation

The solution to this differential equation has a power-series expansion

Using the method of undetermined coefficients obtain a recursion formula relating the terms to the terms . Give an explicit formula for for each and find the sum of the series.

First, we differentiate twice,

From the initial conditions and we have

Now, we plug the expressions for , and back into the given differential equation to get

Then, we substitute in the values we obtained above and to obtain

Since this sum is equal to 0 for all , we know that the coefficients must all be equal to 0. First, we solve for and ,

Then, we obtain the recursion relation,

For the even terms, since we have . For the odd terms we start with and have

Therefore, the remaining odd terms . Therefore, the have the following coefficients

So the sum is