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