Consider a continuous$-$time system

$y^{}(t)+3y^{}(t)+2y(t)=u(t),$

where $y(t)$ is the output and $u(t)$ is the input.

$($a$)$ Describe the transfer function of the system. That is$,$ in case $u(t)=e^{jwt}$ with some winR, describe the output $y(t)$ of the system, with respect to the value of $w.$

$($b$)$ Find the output $y(t),$ when $u(t)=cos(3t).$

$($c$)$ Find the Fourier series $\{c_k{\}}_k=-{}^{}$ for the signal

$x(t)$:$=$rect$(t)**{}_2(t)$

with respect to its period $T=2,$ where the values of rect$(t)$ is $1$ for $-0\mathrm{.}5t0\mathrm{.}5$ and zero for the rest, and ${}_2(t)$ is the delta$-$train signal whose period is $2.$

$($d$)$ We have represented the signal $x(t)$ by

$x(t)={\displaystyle \underset{k=-}{\overset{}{}}}{c}_k{e}^{jkt}$

Now, describe the output $y(t)$ of the system $(1),$ when the input $u(t)$ is given by $u(t)-=x(t).$