 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/13a/c/solution

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

Each spatial mode acquires the factor

exp (\int_{0}^{t} μ (s) d s - D k_{n}^{2} t) = exp ((μ_{0} - D k_{n}^{2}) t + μ_{1} sin t) .

The oscillatory factor

e^{μ_{1} s i n t}

is bounded and does not affect the long-term exponential rate. With two open ends, growth therefore occurs exactly when

μ_{0} > \frac{D π ^{2}}{L ^{2}} .

Only the time average of this spatially uniform fluctuating growth rate controls the asymptotic threshold.

 New to topics? Read the docs here!