Chapter 5.6, Problem 21P

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

Consider the initial value problem

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

Where

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

a) Draw the graph of $g(t)$ on an interval such as $0\le t\le 6\pi$ compare the graph with that of $f(t)$ in Problem 19(a)

b) Find the solution of initial value problem.

c) Let $n=15$ and plot the graph of the solution for $0\le t\le 60$. Describe the solution and explain why it behaves as it does. compare it with the solution of Problem 19.

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