Exercise 9

Find the solution of the initial value problem $$\begin{gathered} 2 y'' + y' + 2y = g(t) \\ y(0) = y'(0) = 0 \end{gathered}$$ where $g(t)$ is defined by $$g(t) = u_5(t) - u_{20}(t) = \begin{cases} 1, & 5 \leq t \lt 20, \\ 0, & 0 \leq t \lt 5 \text{ and } t \geq 20. \end{cases}$$