Chapter 5.7, Problem 16P

Consider the initial value problem

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

Where $f_k(t)=[u_{3-k}(t)-u_{3+k}(t)]/2k,$ with $0<k\le 1$

a) Find the solution $y=\varphi (t,k)$ of the initial value problem.

b) Calculate ${\lim }_{k\to 0}\varphi (t,k)$ from the solution found in part(a).

c) Observe that ${\lim }_{k\to 0}{f}_k(t)=\delta (t-3).$ Find the solution ${\varphi }_0(t)$ of the given initial value problem, with $f_k(t)$ replaced by $\delta (t-3)$. Is it true that ${\varphi }_0(t)={\lim }_{k\to 0}\varphi (t,k)?$

d) Plot $\varphi \left(t,\frac{1}{2}\right)$, $\varphi \left(t,\frac{1}{4}\right)$ and ${\varphi }_0(t)$ on the same axes. describe the relation between $\varphi (t,k)$ and ${\varphi }_0(t)$