II$.[5]$ Consider a mass suspended from a spring attached to a rigid support as shown in Figure $6$ b$.$ Gravity pulls the mass downward and the restoring force of the spring pulls the mass upward. If the mass is displaced from the equilibrium position, it oscillates up and down. This behavior can be modeled by a second$-$order differential equation shown below where $F$ is the external force applied to the spring, y is the displacement of the spring, m is the mass suspended from the spring, c is the damping force, and k is the spring constant.

$m\frac{d^{2}y}{d{t}^2}+c\frac{dy}{dt}+ky=F$

If mass is $4lb,k=16l\frac{b}{f}t,c=32lb-f\frac{t}{sec},$ and $F=0,$ find the displacement $y(t)$ for $t>0.$

Assume $y(0)=1,y^{\text{'}}(0)=10.$