N2003 P2 Q5

It is given that \(y=\sin \left[ \ln \left( 1+x \right) \right]\). Show that

1. \(\left( 1+x \right)\frac{\mathrm{d}y}{\mathrm{d}x}=\cos \left[ \ln \left( 1+x \right) \right]\),[1]
2. \({{\left( 1+x \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+\left( 1+x \right)\frac{\mathrm{d}y}{\mathrm{d}x}+y=0\).[3]

Find the Maclaurin series for \(y\), up to and including the term in \({{x}^{3}}\).[5]