Given that $y={{\mathrm{e}}^{{{\sin }^{-1}}\left( 2x \right)}}$, show that $\left( 1-4{{x}^{2}} \rig

Question

Given that $y={{\mathrm{e}}^{{{\sin }^{-1}}\left( 2x \right)}}$, show that $\left( 1-4{{x}^{2}} \right)\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}-4x\frac{\mathrm{d}y}{\mathrm{d}x}=4y$. 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 ${{\mathrm{e}}^{\frac{\pi }{2}}}$, by substituting a suitable value of $x$ in the Maclaurin series $y$. Suggest one way to improve the accuracy of the approximated value obtained.

Accepted Answer

$y = 1 + 2x + 2x^2 + \frac{8}{3}x^3 + \cdots$; $e^{\pi/2} \approx 2.83$

2006 NJC Promo Q5

2006 NJC Promo Q5

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