 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-6/2/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

S (t) h (x, v) = e^{- t} h (x - t v, v)

. Substituting it into the Duhamel principle gives

f (t, x, v) = e^{- t} f_{0} (x - t v, v) + \int_{0}^{t} e^{- (t - s)} g (s, x - (t - s) v, v) d s .

The spatial shifts all follow the same backward characteristic curve ending at

(x, v)

t

; the velocity is constant.

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