 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/7/solution

The chain rule gives

d (e^{x y}) = e^{x y} (y d x + x d y)

, so the given total differential is exact. Thus

f = e^{x y} + C

; two such functions differ by a constant because their gradient difference vanishes on the connected plane. The assigned value at the origin forces

C = - 1

f (x, y) = e^{x y} - 1 .

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