Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 3 / 17C / Solution

Expanding the order condition about $w=1$ yields $\rho (w)={\sigma }_s{\sum }_{l=1}^s{w}^{s-l}(w-1{)}^l/l$ and ${\sigma }_s=(\sum 1/l{)}^{-1}$. Thus BDF2 has $\rho =w^2-4w/3+1/3$, ${\sigma }_2=2/3$, and BDF3 has $\rho =w^3-18w^2/11+9w/11-2/11$, ${\sigma }_3=6/11$. Their first characteristic polynomials satisfy the root condition, so consistency plus Dahlquist equivalence gives convergence. For BDF2 the stability boundary $z=\rho (e^{i\theta })/({\sigma }_2{e}^{2i\theta })$ has nonnegative real part; hence the whole left half-plane is stable.

Solved by gpt-5.6-sol high.