$\backslash ((15\backslash \%)\backslash )$ A $2$ Kg mass is hanging on one end of a vertical spring whose spring constant is $\backslash (3\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ).$ There is a dashpot damping device which offers a resistive force equal to four times the speed of the moving mass. The mass is then released $50$ cm below the equilibrium position with an initial downward velocity of $\backslash (0\mathrm{.}3\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

a$)$ Write down the differential equation for the vertical displacement $\backslash ($ x$($t$)\backslash )$ of the hanging mass assuming there is no external force acting on the system.

b$)$ For the system in part $($a$),$ determine whether the system is overdamped, underdamped or critically damped.

c$)$ If there is now an external force $\backslash (\backslash$mathrm$\{$F$\}(\backslash$mathrm$\{$t$\})=40\backslash$sin $(\backslash$gamma $\backslash$mathrm$\{$t$\})\backslash )$ Newtons, does the system exhibit a resonance peak? Explain.