 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2014/4/b/i/solution

Dividing by

x^{2}

, which is allowed precisely away from zero, gives

f (x, y) = \frac{1 - 2 x}{x ^{2}} y + \frac{e ^{- 1/ x}}{x ^{2} ( 1 + x ^{2} )}, x \neq = 0.

This function is smooth on each of the two half-planes

x > 0

and

x < 0

, and is affine in the dependent variable

y

y

(1 - 2 x) / x^{2}

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