Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 2 / a / Solution

The characteristic polynomials of a linear multistep method areandFor the order conditions for a linear multistep method, put $\alpha =(a,-1-a,1)$ and $\beta =(-(1+5a)/12,,8(1-a)/12,,(5+a)/12)$. The defectsvanish for $q=0,1,2,3$, whileThereforeIndeed, at $a=-1$ one has $C_4=0$ but $C_5=-4/3\ne 0$.