Problem 2. Consider the single-degree-of-freedom mass-spring-dashpot
$m\ddot{y}(t) + c\dot{y}(t) + ky(t) = u(t)$,
where $\dot{y}(0) = y_0$ and $y(0) = y_0$. Find $\hat{y}(s) = \mathcal{L}\{y(t)\}$ and indicate which part of the solution
is the free response and which part of the solution is the forced response. Your solution
should be expressed in terms of $m$, $c$, $k$, $y_0$, $\dot{y}_0$, and $\hat{u}(s) = \mathcal{L}\{u(t)\}$.
Next, let $m = 1$ kg, $c = 6$ kg/s, $k = 10$ kg/s$^2$, $y_0 = 2$ m, $\dot{y}_0 = 5$ m/s, and $u(t) = 0$ N.
Find the solution $y$ (without using a computational tool). Note that this is the free response
because $u$ is zero.
Hint: $\mathcal{L}\{Ae^{-at}\cos\omega t + Be^{-at}\sin\omega t\} = \frac{A(s+a) + B\omega}{(s+a)^2 + \omega^2}$
