In this exercise we will use the Laplace transform to solve the following initial value problem:
y'' + 4y = \begin{cases} 4, & 0 \le t < 2 \\ 0, & 2 \le t \end{cases}, y(0) = -1, y'(0) = -4
(1) First, using $Y$ for the Laplace transform of $y(t)$, i.e., $Y = \mathcal{L}\{y(t)\}$, find the equation obtained by taking the Laplace transform of the initial value problem
(2) Next solve for $Y = $
(3) Finally apply the inverse Laplace transform to find $y(t)$
y(t) =