10. \begin{figure}[h] \end{figure} The motion of a pendulum, shown in Figure 3, is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ where \(\theta\) is the angle, in radians, that the pendulum makes with the downward vertical, \(t\) seconds after it begins to move.

1. 1. Show that a particular solution of the differential equation is $$\theta = \frac { 1 } { 12 } t \sin 3 t$$
   2. Hence, find the general solution of the differential equation. Initially, the pendulum
      • makes an angle of \(\frac { \pi } { 3 }\) radians with the downward vertical
2. is at rest
Given that, 10 seconds after it begins to move, the pendulum makes an angle of \(\alpha\) radians with the downward vertical,
3. determine, according to the model, the value of \(\alpha\) to 3 significant figures. Given that the true value of \(\alpha\) is 0.62
4. evaluate the model. The differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 9 \theta = \frac { 1 } { 2 } \cos 3 t$$ models the motion of the pendulum as moving with forced harmonic motion.
5. Refine the differential equation so that the motion of the pendulum is simple harmonic motion.