$($c$)$ Consider the system described by the same differential equation $0\mathrm{.}5y^{}(t)+y(t)=2u(t),$ but the input is

NOT necessarily a unit step function. Let $U(s)$ and $Y(s)$ be the Laplace transform of $u(t)$ and $y(t),$

respectively. Then the input$/$output relationship of the system can be described by the algebraic equation

$Y(s)=G(s)U(s),$ where $G(s)$ is called the transfer function. Find the transfer function $G(s).(20\%)$

$($d$)$ Let $G(j2)=A{e}^j,$ Compute the magnitude A and the phase $.(20\%)$

$($e$)$ Let $y(0)=0$ and $u(t)=cos2t,$ when $t0.$ Find $y_{ss}(t),$ the output of the system at steady state using the

approach described in Corollary $2\mathrm{.}33.(20\%)$