 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-1/13c/b/solution

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

Extend the forcing oddly across the boundary:

f_{odd} (y, s) = {f (y, s), - f (- y, s), y \geq 0, y < 0.

sin x

is already odd, so its homogeneous evolution on the line is

sin x cos (c t)

. Applying Duhamel's principle to the odd extension gives

u (x, t) = sin x cos (c t) + \frac{1}{2 c} \int_{0}^{t} \int_{x - c (t - s)}^{x + c (t - s)} f_{odd} (y, s) d y d s .

The odd-reflection method makes

u (0, t) = 0

. At

t = 0

the double integral vanishes together with its first time derivative, so the prescribed initial displacement and velocity are also satisfied.

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