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

Question

It is given that $y={{\mathrm{e}}^{2x}}\cos x$. Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=4\frac{\mathrm{d}y}{\mathrm{d}x}-5y$.[3] Find the Maclaurin series for $y$ up to the term in ${{x}^{2}}$.[2] Hence, show that the Maclaurin series for ${{\mathrm{e}}^{x}}\sqrt{\cos x}$ is $1+x+\frac{1}{4}{{x}^{2}}$, up to the term in ${{x}^{2}}$.[2]

Accepted Answer

(b) $y = 1 + 2x + \frac{3}{2}x^2 + \ldots$ (c) $e^x\sqrt{\cos x} \approx 1 + x + \frac{1}{4}x^2$

2022 EJC Promo Q2

2022 EJC Promo Q2

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