$(15$ pts$)$ A mechanical system is shown below. The output of the system is the displacemen y$($t$)$ of the mass, the input of the system is the displacement of x$($t$)$ acting on the damper. Where k $=100$N$/$m$,$ m $=4$ kg$,$b $=10$Ns$/$m

$1)$ Derive the equation of motion of the system

$2)$ Obtain the transfer function of Y$($s$)/$X$($s$).$

$3)$ Obtain the response of the system with a unit step input, assuming zero initial conditi

y

k