 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/1/a/solution

After division by

x^{2}

y^{'} - \frac{y}{x ^{2}} = \frac{e ^{- 1/ x}}{x ^{2}} .

e^{1/ x}

. Thus

\frac{d}{d x} (e^{1/ x} y) = \frac{1}{x ^{2}}, e^{1/ x} y = - \frac{1}{x} + C .

y (1) = 1

gives

C = e + 1

, so

y (x) = (e + 1 - \frac{1}{x}) e^{- 1/ x}, x > 0.

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