 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-3/13c/b/solution

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

Since

f (0) = 0

and

f^{'} (0) = - r

f (u) = - r u + O (u^{2}) .

For a spatial Fourier mode of the perturbation, part a gives

\frac{\partial u}{\partial t} = [- D k^{2} - r + ρ (1 - \frac{1}{1 + k ^{2}})] u .

Thus

u (k, t) \propto e^{σ (k) t}

σ (k) = - D k^{2} - r + ρ \frac{k ^{2}}{1 + k ^{2}} .

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