Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 17D / b / Solution

A differential equation is stiff when it contains rapidly decaying modes on time scales much shorter than those of interest, forcing an explicit method to take very small steps for stability rather than accuracy.

Hereso the decay rates are $1$ and $100$. For a negative real eigenvalue, forward Euler requirestherefore the fast mode imposesFor backward Euler the amplification factors arewhose moduli are at most one for every $h\ge 0$. ThusThis is the stiff two-mode linear system.

Solved by gpt-5.6-sol high.