2024 RI Promo Q6

It is given that \(\mathrm{e}^y = 1 + \sin 3x\).

1. Show that
   \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.\)
   Hence find the series expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying your answer.[5]
2. Using standard series from the List of Formulae, verify that the series expansion obtained in part (a) is correct.[3]