It is given that $y={{\mathrm{e}}^{-x}}\sin x+x-1$. Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}

Question

It is given that $y={{\mathrm{e}}^{-x}}\sin x+x-1$. Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=k{{\mathrm{e}}^{-x}}\cos x$, where $k$ is a constant to be determined. [2] By further differentiation of this result, find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$. [3] By using the result in part (b) and standard series from the List of Formulae (MF26), find the expansion of $\frac{{{\mathrm{e}}^{-x}}\sin x+x-1}{\cos 2x}$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$, giving the coefficients in exact form. [3]

Accepted Answer

$k = -2$; $y \approx -1 + 2x - 3x^2 + \dfrac{13}{3}x^3$

2023 CJC P2 Q2

2023 CJC P2 Q2

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