 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/14/a/i/solution

P_{y} = Q_{x}

P_{y} = 2 y sin (a x) + 2 x y cos (a x), Q_{x} = 2 y sin (a x) + 2 a x y cos (a x) .

Thus

(1 - a) x y cos (a x) = 0

must hold identically. Since

x y cos (a x)

is not identically zero for a fixed nonzero

a

, the only possibility is

a = 1.

The explicit potential in part (ii) verifies sufficiency.

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