ACJC Maclaurin Series Tutorial Q5

Given that \(y={{\left( {{\sin }^{-1}}x \right)}^{2}}\), prove that \(\left( 1-{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-x\frac{\mathrm{d}y}{\mathrm{d}x}=2\).

By repeated differentiation of this result, and using Maclaurin series, find the first two non-zero terms in the expansion of \(y\) in ascending powers of \(x\).

By using the standard series expansions for \(\ln (1+x)\) and \({{\mathrm{e}}^{x}}\), show that, when \(x\) is small enough for powers of \(x\) above fourth to be neglected,\({{\left( {{\sin }^{-1}}x \right)}^{2}}=\frac{1}{6}\left( \ln (1+{{x}^{2}})+5{{\mathrm{e}}^{{{x}^{2}}}}-5 \right)\).

Hence find an approximate value for \({{\pi }^{2}}-30{{\mathrm{e}}^{\frac{1}{4}}}\), giving your answer in the form \(c\ln \frac{5}{4}+d\), where \(c\) and \(d\) are integers.