RI Maclaurin Series Tutorial Q3

It is given that \(y=\sin \left[ \ln \left( 1+2x \right) \right]\).

1. Show that \({{\left( 1+2x \right)}^{2}}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+2\left( 1+2x \right)\frac{\mathrm{d}y}{\mathrm{d}x}+ky=0\), where \(k\) is a constant to be found.
   Hence, find the first three non-zero terms of the Maclaurin expansion for \(y\).[6]
2. Using standard series from the List of Formulae, verify the correctness of your result from part (i) up to and including the term in \({{x}^{3}}\).[2]
   Explain why the expansion is not valid when \(x=-\frac{1}{2}\).[1]