Consider the function defined by the power series

Determine the interval of convergence for the power series and show that satisfies the differential equation

First, we use the ratio test to determine the interval of convergence,

Hence, the series converges for all . Then, to show that it satisfies the given differential equation we take the first two derivatives,

Then we have

Therefore, indeed is a solution of the given differential equation.

Now, to find the sum we first need to get the general form of the solutions for the differential equation

First, we find the general form of the solutions of the homogeneous equation

In this case we have an equation of the form where and . From this we can compute and . Therefore, the general form of the solutions is

Then, we can find a particular solution of the given equation by inspection since

is a solution. Therefore, the general solution to the given inhomogeneous equation is

Now, in the particular case we also have the initial condition and so we have

Furthermore, since is an odd function we must have

Therefore, we conclude and . And so,