 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ii/paper-2/38a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-05

f

has expansion

Γ_{5} f = h^{2} Δ f + \frac{h ^{4}}{12} (f_{xxxx} + f_{yyyy}) + O (h^{6}) .

The corrected right side

h^{2} f + (h^{2} /12) Γ_{5} f

cancels the

h^{4} Δ f /12

term in

Γ_{9} u

. Consequently

h^{- 2} (Γ_{9} u - h^{2} f - \frac{h ^{2}}{12} Γ_{5} f) = O (h^{4}) .

More precisely, the unscaled residual's

h^{6}

- (u_{xxxxxx} + u_{yyyyyy}) /240 + (u_{xxxx yy} + u_{xx yyyy}) /144

f

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