Chapter 5.6, Problem 18P

Consider the initial value problem

$y\text{'}\text{'}+\frac{1}{3}y\text{'}+4y=f_k(t),$ $y(0)=0,$ $y\text{'}(0)=0.$

Where

$f_k(t)=\{\begin{array}{c}1/2k,\\ 0,\end{array}\begin{array}{c}4-k\le t<4+k\\ 0\le t<4-k\end{array}\begin{array}{c}\\ \text{and}\end{array}\begin{array}{c}\\ t\ge 4+k\end{array}$

and $0<k<4$

a) Sketch the graph of $f_k(t).$ Observe that the area under the graph is independent of $k$. If $f_k(t)$ represent a force, this means that the product of the magnitude of the force and the time interval during which it acts does not depend on $k$

b) Write $f_k(t).$ in terms of the unit steps function and then solve the given initial value problem

c) Plot the solution for $k=2,k=1,$ and $k=\frac{1}{2}$. Describe how the solution depends on $k.$