Consider the function defined by the power series
Determine the interval of convergence for and show that it satisfies the differential equation
Find and .
First, to determine the interval of convergence for we use the ratio test,
Therefore, the series converges for all . Next, to show that it satisfies the given differential equation, we take the first two derivatives,
Then, we have the differential equation ,
But, for this equation to hold we must have (since there is no constant term in on the left) and we must also have since there the coefficient of on the left is 1 and the only possible term on the right is if . Using these values of and we verify that the given differential equation is satisfied since we have
Hence, we indeed have .