A mass of $5$ kg stretches a spring $10$ cm$.$ The mass is acted on by an external force of $10*$sin$($t$/2)$ N $($newtons$)$ and moves in a medium that imparts a viscous force of $2$ N when the speed of the mass is $4$ cm$/$sec$.$ If the mass is set in motion from its equilibrium position with an initial velocity of $3$ cm$/$sec$,$ formulate the initial value describing the motion of the mass and then solve the initial value problem.