4. A particle $P$ of mass 9 kg moves along the horizontal positive $x$-axis under the action of a force directed towards the origin. At time $t$ seconds, the displacement of $P$ from $O$ is $x$ metres, $P$ is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the force has magnitude $16 x$ newtons. The particle $P$ is also subject to a resistive force of magnitude $24 v$ newtons.
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of motion of $P$ is

$$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 24 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 16 x = 0$$

It is given that the general solution of this differential equation is

$$x = \mathrm { e } ^ { - \frac { 4 } { 3 } t } ( A t + B )$$

where $A$ and $B$ are arbitrary constants.\\
When $t = \frac { 3 } { 4 } , P$ is travelling towards $O$ with its maximum speed of $8 \mathrm { e } ^ { - 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $x = d$.
\item Find the value of $d$.
\item Find the value of $x$ when $t = 0$
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2016 Q4 [12]}}