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An Impulsive Forcing

DIFFEQ-LSHE72

Sarah was solving a linear system with a periodic signal and an impulse forcing. Suppose the system can be modeled with:

$$y''+2y'+3y=\cos(t)+\delta(t-\pi)$$

...and the initial conditions are given by $y(0)=0, y'(0)=0$.

Sarah wanted to find $y(t)$ by taking the Laplace Transform. Suppose Sarah found the function $y(t)$ successfully.

Then, the function she found could be (note that $u(t)$ is the Heaviside step function and $u_{\pi}=u(t-\pi)$):

A

$-\cfrac{1}{4}e^{-t}\cos(\sqrt{2}t)-\cfrac{1}{4}e^{-t}\sin(\sqrt{2}t) +\cfrac{1}{4}\cos t+\cfrac{1}{4}\sin t+\cfrac{1}{2}u_{\pi}e^{-(t-\pi)}\sin(\sqrt{2}(t-\pi))$

B

$-\cfrac{1}{4}e^{-t}\cos(\sqrt{2}t)+\cfrac{1}{4}\cos t+\cfrac{\sqrt{2}}{2}u_{\pi}e^{-(t-\pi)}\sin(\sqrt{2}(t-\pi))$

C

$-\cfrac{1}{4}e^{-t}\cos(\sqrt{2}t)-\cfrac{\sqrt{2}}{4}e^{-t}\sin(\sqrt{2}t) +\cfrac{1}{4}\cos t+\cfrac{1}{4}\sin t+\cfrac{\sqrt{2}}{2}u_{\pi}e^{-(t-\pi)}\sin(\sqrt{2}(t-\pi))$

D

None of the above