Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 17C / a / Solution

Insert the exact data $y(t)=P(t)$ into the linear multistep method. Its local residual is the linear functionalTaylor expansion about $t_n$ shows that a method has order $p$ exactly when this functional annihilates the monomials $1,t,\dots ,t^p$; by linearity this is equivalent to annihilating every polynomial of degree at most $p$. This is precisely