2006 NJC Promo Q5
Timothy Gan
2006 NJC Promo Q5
Given that $y={{\text{e}}^{{{\sin }^{-1}}\left( 2x \right)}}$, show that $\left( 1-4{{x}^{2}} \right)\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}-4x\frac{\text{d}y}{\text{d}x}=4y$.
By further differentiating this result, find the Maclaurin series for $y$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.
Hence, find an approximation value of ${{\text{e}}^{\frac{\pi }{2}}}$, by substituting a suitable value of $x$ in the Maclaurin series $y$.
Suggest one way to improve the accuracy of the approximated value obtained.
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