Chapter 5.6, Problem 19P

In Problem 19 through 23, we explore the effect of some nonsinusoidal periodic forcing functions :

Consider the initial value problem

$y\text{'}\text{'}+y=f(t),$ $y(0)=0,$ $y\text{'}(0)=0.$

Where

$f(t)=1+2{\displaystyle \underset{k=1}{\overset{n}{\sum }}{(-1)}^k{u}_{k\pi }(t)}$

a) Find the solution of initial value problem.

b) Let $n=15$ and plot the graph of $f(t)$ and $y(t)$ for $0\le t\le 60$ on the same set of coordinate axes. Describe the solution and explain why it behaves as it does.

c) Investigate how the solution changes as $n$ increases. What happens as $n\to \infty ?$