<p>It is given that&nbsp;\(\mathrm{e}^y = 1 + \sin 3x\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">\(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.\) </div><div class="list-item-paragraph">Hence find the series expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying your answer.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Using standard series from the List of Formulae, verify that the series expansion obtained in <strong>part (a)</strong> is correct.<span class="inline-marks">[3]</span></div></li></ol>

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

\(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 + 9 = 9\mathrm{e}^{-y}.\) 

Hence find the series expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying your answer.

Using standard series from the List of Formulae, verify that the series expansion obtained in **part (a)** is correct.

2024 RI Promo Q6

2024 RI Promo Q6