2021 NYJC Promo Q7
It is given that $y=\sqrt{1+\ln \left( 1+\sin 2x \right)}$.
(i)
Show that $y\frac{\text{d}y}{\text{d}x}=\frac{\cos 2x}{1+\sin 2x}$.
[1]
(ii)
Show that $y\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+{{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=\frac{k}{\left( 1+\sin 2x \right)}$, where $k$ is a constant to be determined.
[3]
(iii)
Hence show that the Maclaurin series of $y$ is $1+x-\frac{3}{2}{{x}^{2}}+\frac{13}{6}{{x}^{3}}+…$.
[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 $\ln \left( 1+x \right)$ 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]
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