 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/17c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Insert the exact data

y (t) = P (t)

L_{h} [P] = \sum_{k = 0}^{s} ρ_{k} P (t_{n + k}) - h \sum_{k = 0}^{s} σ_{k} P^{'} (t_{n + k}) .

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

k = 0 \sum s ρ_{k} P (t_{n + k}) = h k = 0 \sum s σ_{k} P^{'} (t_{n + k}) .

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