$2y''+y'+2y=\delta(t-5)$
$y(0)=0, y'(0)=0$
Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to rest again after one cycle.
Determine the impulse $k\delta(t-t_{0})$ that should be applied to system in order to accomplish this objective (bring sys to rest).
Now how can I calculate the extra impulse to applied at time $T$? I have no idea. I added an extra impulse to the system and tried to set $y(t)=0$ at that time $T$ but no luck. How can I kill a sinusoidal oscillation by applying an extra impulse?
Books answer: $-\exp(-T/4)\delta(t-5-T)$