$(3$ pts$)($Canonical second order system$)$ Consider the usual mass$-$spring system

$m{x}^{}+c{x}^{}+kx=f$

where $x(t)$ is the displacement of the spring and $f(t)$ is the applied force, and the parameters $m,c,k$ carry their usual meanings.

$($a$)(2$ pts$)$ Show that by rescaling, we can write the dynamics $(1)$ in the canonical form

$x^{}+2{}_n{x}^{}+{}_n^{2}x={}_n^{2}u$

where $u(t)=f\frac{t}{k}.$ Express the natural frequency ${}_n$ and the damping ratio $$ in terms of the parameters $m,c,k.$

$($b$)(1$ pts$)$ Find the transfer function $G(s)=\frac{x(s)}{U(s)}.$ Observe that this has the same form of the canonical second order system as covered in class.