4. [2360071423 (33 pts)] A 2-kg mass is attached to spring with restoring/spring constant of $2 Nt / m$ . The apparatus is aligned horizontally with a damping constant of $5 N / m / \sec$ , and is forced by $f (t) = 3 e^{- t} + 4 Nt$ . Initially, $x (0) = - 4$ and $\dot{x} (0) = - 3$ .
(a) (2 pts) Where is the mass with respect to its equilibrium position when $t = 0$ and in what direction is it moving at that time?
(b) (3 pts) Is the oscillator over-, under-, or critically damped? Justify your answer.
(c) (3 pts) Is the oscillator in resonance? Justify your answer.
(d) (15 pts) Find the position of the mass at any time $t$ , that is, solve an appropriate initial value problem.
(e) (2 pts) From your answer to part (d), identify the transient and steady state solutions.
(f) $(8 pts)$ Write the initial value problem from part (d) as a system of differential equations/IVPs, using matrices and vectors in your answer.

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Final Answer: The mass is $4$ units to the left of its equilibrium position when $t = 0$ and it is moving to the $l e f t$ at that time.

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Step 1 ：The equilibrium position of a mass-spring system is the position where the spring force balances the gravitational force, which is usually defined as $x = 0$ .

Step 2 ：The initial position of the mass is given as $x (0) = - 4$ , which means the mass is 4 units to the left of the equilibrium position at $t = 0$ .

Step 3 ：The initial velocity of the mass is given as $\dot{x} (0) = - 3$ , which means the mass is moving to the left at $t = 0$ (since negative velocity indicates leftward motion in this context).

Step 4 ：Final Answer: The mass is $4$ units to the left of its equilibrium position when $t = 0$ and it is moving to the $l e f t$ at that time.

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