Problem $4.$ Consider the spring$-$mass system shown in the

figure. The displacement of the mass $m$ as a function of time

is as follows:

$x=x_{0}cos({}_{n}t)+\frac{x_0^{}}{{}_n}cos({}_{n}t)$

where $x_0$ is the initial displacement equals to $0\mathrm{.}1m,x_0^{}$ is the initial velocity equals to $1\frac{m}{s},$

and ${}_n$ is the natural frequency of the system equals to $4ra\frac{d}{s}.$

Calculate the acceleration $($second time derivative of displacement$)$ of the mass after $1$ s

with a time step size of $0\mathrm{.}1$ s and $0\mathrm{.}2$ s using the centered finite$-$divided difference. Determine

the true percent relative error for both time steps. Comment on the accuracy of your results.