A particle of mass $m$ kg is attached to one end of a light elastic string of natural length $a$ metres and modulus of elasticity $5ma$ newtons. The other end of the string is attached to a fixed point $O$ on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length $2a$ metres and then released at time $t = 0$. During the subsequent motion, when the particle is moving with speed $v$ m s$^{-1}$, the particle experiences a resistance of magnitude $4mv$ newtons. At time $t$ seconds after the particle is released, the length of the string is $(a + x)$ metres, where $0 \leqslant x \leqslant a$.

\begin{enumerate}[label=(\alph*)]
\item Show that, from $t = 0$ until the string becomes slack,
$$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
\item Hence express $x$ in terms of $a$ and $t$. [6]
\item Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form $ka$, where $k$ is a constant to be found correct to 2 significant figures. [4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2014 Q6 [13]}}