2021 NYJC Promo Q7

It is given that \(y=\sqrt{1+\ln \left( 1+\sin 2x \right)}\).

1. Show that \(y\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\cos 2x}{1+\sin 2x}\).[1]
2. Show that \(y\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}=\frac{k}{\left( 1+\sin 2x \right)}\), where \(k\) is a constant to be determined.[3]
3. Hence show that the Maclaurin series of \(y\) is \(1+x-\frac{3}{2}{{x}^{2}}+\frac{13}{6}{{x}^{3}}+...\).[3]
4. Expand \({{\left( 1+x-\frac{3}{2}{{x}^{2}}+\frac{13}{6}{{x}^{3}} \right)}^{2}}\) in powers of \(x\) up to and including the term in \({{x}^{3}}\), simplifying the coefficients. By using the standard series expansions of \(\sin x\) and \(\ln \left( 1+x \right)\) from the List of Formulae (MF26), explain briefly how the result can be used as a check on the correctness of the first four terms in the series for \(y\).[3]