2021 YIJC P1 Q8

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

1. Show that \({{\left( 1+\mathrm{e}x \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+\mathrm{e}\left( 1+\mathrm{e}x \right)\frac{\mathrm{d}y}{\mathrm{d}x}=-{{\mathrm{e}}^{2}}y\).[2]
2. By further differentiation of the result in (i), find the Maclaurin series for \(y\), up to and including the term in \({{x}^{3}}\).[4]
3. By using appropriate standard series expansions from the List of Formulae (MF26), verify the correctness of the Maclaurin series for \(y=\sin \left( \ln \left( 1+\mathrm{e}x \right) \right)\) found in part (ii).[2]