 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-6/1/c/solution

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

For this force,

\dot{V} = V

, hence

V (t) = e^{t} v

and

X (t) = x + (e^{t} - 1) v

. Invert the characteristic flow map: the initial velocity for an endpoint

(x, v)

e^{- t} v

, and the initial position is

x - (1 - e^{- t}) v

. Constancy along each characteristic curve gives

f (t, x, v) = f_{0} (x - (1 - e^{- t}) v, e^{- t} v) .

There is no amplitude factor here: this is scalar advection, not the conservative continuity equation for a compressible density.

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