Consider the function f(x) defined by the power series,
Determine the interval of convergence of f(x) and show that f(x) satisfies the differential equation
First, to determine the radius of convergence we use the ratio test,
Therefore, the series converges for all (or the radius of convergence is ). Next, to show that satisfies the given differential equation we take the first two derivatives,
Plugging this into the given differential equation we have
Hence, indeed satisfies the given differential equation.