2020 TMJC Promo Q6
Do not use a calculator for this question
(a)
Give that $y={{\text{e}}^{\frac{2}{x+1}}}$, where $x\ne -1$, show that
${{(x+1)}^{2}}\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+2(x+2)\frac{\text{d}y}{\text{d}x}=0$.
By further differentiation of this result, or otherwise, find the Maclaurin series of $y$, up to and including the term in ${{x}^{3}}$.
Leave the coefficients in exact form.
[7]
(b)
Using appropriate expansions from the List of Formulae (MF26), find the expansion of ${{(4-x)}^{-1}}\cos \,\,3x$, up to and including the term in ${{x}^{3}}$.
Give the coefficients as exact fractions in their simplest form.
[5]
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