2017 CJC Promo Q9

It is given that \(y=\frac{\cos x}{1-2x}\).

1. Show that \(\left( 1-2x \right)\left( \frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+y \right)-4\frac{\mathrm{d}y}{\mathrm{d}x}=0\).[3]
2. By repeated differentiation of the result in part (i), find the Maclaurin series for \(y\), in ascending powers of \(x\), up to and including the term in \({{x}^{3}}\).[3]
3. Suppose that \(x\) is sufficiently small for \({{x}^{3}}\) and higher powers of \(x\) to be neglected, verify the correctness of the first three terms of the series obtained in part (ii) using the appropriate expansions from the List of Formulae (MF26).[3]