$(2)$ Suppose a mass$-$spring$-$damper system has mass of $\frac{1}{2}kg,$ dashpot term $c=3,$ and spring stiffness $k=4.$ Assume the initial position and initial velocity are $x_0=2$ and $v_0=0,$ respectively.

$($a$)$ Write and solve the differential equation to get an expression for the position, $x(t).(4$pts$)$

$($b$)$ Now assume that $k=6.$ Solve the differential equation for $x(t).(4$pts$)$

Problems from Sec. $3\mathrm{.}5$