Problem $5$: For the mechanical system shown,

$($i$)$ derive the transfer functions $\frac{x_3(s)}{x_1(s)}.$

$($ii$)$ find a state$-$space representation for the system.

$($iii$)$ using $G(s)=C(sI-A{)}^{-1}B+D,$ verify that you can get the same TF from the state$-$space representation $($show your workings$).$

$($iv$)$ find the poles of the system.