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$y(0)=0, y'(0)=0$

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 sinusoidal oscillation by applying an extra impulse?

Books answer: $-\exp(-T/4)\delta(t-5-T)$

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