Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 6C / c / Solution

With hostile Dirichlet boundaries, expand in $\sin (k\pi x/L)$. The $k$th amplitude grows at rate $\alpha -D(k\pi /L{)}^2$. Persistence requires the principal rate to be nonnegative, hence $\alpha \ge D{\pi }^2/L^2$; strict inequality gives growth.

Solved by gpt-5.6-sol high.