Chapter 5.6, Problem 17P

Modify the problem in Example 1 of this section by replacing the given forcing function g(t) by

$f(t)=\frac{u_5(t)(t-5)-u_{5+k}(t)(t-5-k)}{k}.$

a) Sketch the graph of $f(t)$ and describe how it depends on $k$. For what value of $k$ is $f(t)$ identical to $g(t)$ in the example ?

b) Solve the initial value problem

$y\text{'}\text{'}+4y=f(t),$ $y(0)=0,$ $y\text{'}(0)=0$

c) The solution in part (b) depends on $k$, but for sufficiently large $t$ the solution is always a simple harmonic oscillation about $y=\frac{1}{4}$. Try to decide how the amplitude of this eventual oscillation depends on $k$. Then confirm your conclusion by plotting the solution for a few different values of $k.$