NYJC/2021/JC1/Promo/P1/Q07

It is given that $y=\sqrt{1+\ln (1+\sin 2 x)}$.
(i) Show that $y \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\cos 2 x}{1+\sin 2 x}$.

[1]

(ii) Show that $y \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2=\frac{k}{(1+\sin 2 x)}$, where $k$ is a constant to be determined.

[3]

(iii) Hence show that the Maclaurin series of $y$ is $1+x-\frac{3}{2} x^2+\frac{13}{6} x^3+\ldots$

[3]

(iv) 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 $\ln (1+x)$ 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]