(1 point) Consider the initial value problem
$my'' + cy' + ky = F(t)$, $y(0) = 0$, $y'(0) = 0$
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 $F(t) = 30 \sin(6t)$ Newtons.
a. Solve the initial value problem.
$y(t) = $
b. Determine the long-term behavior of the system. Is $\lim_{t \to \infty} y(t) = 0$? 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 $t$, $y(t) \approx$