 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/38c/e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Put

z = x - c t

and

h (x, t) = f (z)

- c f^{'} + \frac{d}{d z} [\frac{ρ g f ^{3}}{3 μ} (sin α - cos α f^{'})] = 0.

One integration gives the required first-order autonomous equation

\frac{ρ g f ^{3}}{3 μ} (sin α - cos α f^{'}) - c f = C,

where

C

is a constant fixed by the far-field film heights or other boundary data.

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