 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-4/6b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Integrating the population equation over the line and using vanishing boundary flux gives

\dot{M} = - μ M

, where

M (t) = \int n (x, t) d x

. Since

M (0) = Q

, one has

M (t) = Q e^{- μ t}

. On the other hand the similarity substitution

x = λ (t) η

gives

M (t) = e^{- μ t} \int_{R} f (η) d η .

Consequently

\int_{R} f (η) d η = Q .

The decay condition is on the nonlinear diffusive flux

n n_{x}

; compact support also satisfies it.

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