Moderate# 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 $u_{\pi}=u(t-\pi)$):