Home » Blog » Determine the interval of convergence of a given power series and show that it satisfies a differential equation

# Determine the interval of convergence of a given power series and show that it satisfies a differential equation

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, 