Consider the initial value problem

$$m y^{\,\prime\prime} + c y^{\,\prime} + k y = F(t), \ \ \ y(0) = 0, \ \ \ y^{\,\prime}(0) = 0$$
modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force $F(t)$, where the unit of force is the Newton (N). Assume that $m = 2$ kilograms, $c = 8$ kilograms per second, $k = 80$ Newtons per meter, and the applied force in Newtons is

$$F(t) = \left\lbrace \begin{array}{lcl} 20 && \mbox{if } 0 \leq t \leq \pi/2, \\ 0 && \mbox{if } t > \pi/2. \end{array} \right.$$

1. Solve the initial value problem, using that the displacement $y(t)$ and velocity $y^{\,\prime}(t)$ remain continuous when the applied force is discontinuous.
   
   For $0 \leq t \leq \pi/2$,   $y(t) =$ help (formulas)
   
   For $t > \pi/2$,   $y(t) =$ help (formulas)
   
   
2. Determine the long-term behavior of the system. Is $\displaystyle \lim_{t\to\infty} y(t) = 0$? If it is, enter zero. If not, enter a function that approximates $y(t)$ for very large positive values of $t$.
   
   For very large positive values of $t$, $y(t) \approx$ help (formulas)