Consider the function defined by the power series

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

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

Hence, the series converges for all .

To show that it satisfies the given differential equation, we first take the derivative,

Then, it satisfies the given differential equation since

Then, since the given differential equation is a first-order linear differential equation of the form

with we know that the solutions are uniquely determined by the formula

Since we have the initial condition (by plugging in to the power series expansion for ) we have and the unique solution of this differential equation is