It is given that $y=\sqrt{{{\mathrm{e}}^{x}}\cos x}$. Show that $2y\frac{\mathrm{d}y}{\mathrm{d}x}={

Question

It is given that $y=\sqrt{{{\mathrm{e}}^{x}}\cos x}$. Show that $2y\frac{\mathrm{d}y}{\mathrm{d}x}={{y}^{2}}-{{\mathrm{e}}^{x}}\sin x$. [2] By further differentiation of the result in part (i), find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$. [4] Using the standard series from the List of Formulae (MF 26). Expand $\sqrt{{{\mathrm{e}}^{x}}\cos x}$ as far as the term in ${{x}^{2}}$ and verify that the same result is obtained in part (ii). [3]

Accepted Answer

(i) $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2}$ (ii) $y = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + ...$ (iii) Verified using standard series

2020 MI P1 Q6

2020 MI P1 Q6

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