$4.$ An exercise machine for the arm can be described as the mechanical system below.

a$)$ Write the second order differential equation describing the system.

b$)$ Write the Laplace transform describing the impulse response of the system.

c$)$ If $\backslash ($ M$=1\backslash$mathrm$\{$~kg$\},$ B$=5\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash )$ and $\backslash ($ K$=4\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ),$ determine the transfer function response of the system as the sum of two polynomials.

d$)$ Determine the impulse response of the system in the time domain.