Sketch or describe solution behavior

6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = \sin 2 t$$ Describe the behaviour of \(x\) as \(t \rightarrow \infty\), justifying your answer.

The solution of a second order differential equation is \(f(t)\) The differential equation models heavy damping. Which one of the statements below could be true? Tick \((\checkmark)\) one box. [1 mark] \(f(t) = 2\mathrm{e}^{-t} \cos(3t) + 5\mathrm{e}^{-t} \sin(3t) \quad \square\) \(f(t) = 3\mathrm{e}^{-t} + 4t\mathrm{e}^{-t} \quad \square\) \(f(t) = 7\mathrm{e}^{-t} + 2\mathrm{e}^{-2t} \quad \square\) \(f(t) = 8\mathrm{e}^{-t} \cos(3t - 0.1) \quad \square\)