Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 39A / b

Order $p$ applied to this particular differential equation forces the local stability polynomial to agree with the exact exponential through degree $p$:As its degree is at most $s$, the nonzero coefficient $1/p!$ cannot be present for $p>s$. ThereforeThe implication is a necessary stability-polynomial condition; it does not assert that every such polynomial automatically gives the full nonlinear order conditions.