It is given that $\frac{\mathrm{d}y}{\mathrm{d}x}+xy={{\mathrm{e}}^{-x}}$ and $y=1$ when $x=0$. Find

Question

It is given that $\frac{\mathrm{d}y}{\mathrm{d}x}+xy={{\mathrm{e}}^{-x}}$ and $y=1$ when $x=0$. Find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$. [5] Hence, or otherwise, state the first three non-zero terms of the Maclaurin series for $\frac{\mathrm{d}y}{\mathrm{d}x}$. [1]

Accepted Answer

(i) $y=1+x-{{x}^{2}}-\frac{1}{6}{{x}^{3}}+...$ (ii) $\frac{\mathrm{d}y}{\mathrm{d}x}=1-2x-\frac{1}{2}{{x}^{2}}+...$

2020 VJC Promo Q1

2020 VJC Promo Q1

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