Chapter 5.7, Problem 15P

Consider the initial value problem

$y\text{'}\text{'}+\gamma y\text{'}+y=k\delta (t-1),$ $y(0)=0,$ $y\text{'}(0)=0,$

Where $\left|k\right|$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coeffiecnt (or resistance).

a) Let $\gamma =\frac{1}{5}$. Find the value of $k$ for which the response has a peak value of $6$; call this value $k_1$.

b) Repeat part (a) for $\gamma =\frac{1}{10}$

c) Determine how $k_1$ very as $\gamma$ dreacreses. what are the value of $k_1$ when $\gamma =0?$