Answer only one of the following two alternatives.

EITHER

The variables \(z\) and \(x\) are related by the differential equation
\(3 z^{2} \frac{\mathrm{~d}^{2} z}{\mathrm{~d} x^{2}}+6 z^{2} \frac{\mathrm{~d} z}{\mathrm{~d} x}+6 z\left(\frac{\mathrm{~d} z}{\mathrm{~d} x}\right)^{2}+5 z^{3}=5 x+2 .\)

Use the substitution \(y=z^{3}\) to show that \(y\) and \(x\) are related by the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+2 \frac{\mathrm{~d} y}{\mathrm{~d} x}+5 y=5 x+2 .\)

Given that \(z=1\) and \(\frac{\mathrm{d} z}{\mathrm{~d} x}=-\frac{2}{3}\) when \(x=0\), find \(z\) in terms of \(x\).

Deduce that, for large positive values of \(x, z \approx x^{\frac{1}{3}}\).

OR

The curve \(C\) has equation
\(y=\frac{x(x+1)}{(x-1)^{2}} .\)
(i) Obtain the equations of the asymptotes of \(C\).

(ii) Show that there is exactly one point of intersection of \(C\) with the asymptotes and find its coordinates.

(iii) Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) and hence
(a) find the coordinates of any stationary points of \(C\),
(b) state the set of values of \(x\) for which the gradient of \(C\) is negative.

(iv) Draw a sketch of \(C\).