 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-2/17a/a/solution

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

Write the internal stage as

k_{2} = f (y_{n} + h [(1 - θ) k_{1} + θ k_{2}]) .

The two methods have

θ = 1/2

and

3/4

. Expansion gives stability function

R_{θ} (z) = \frac{1 + ( 1 - θ ) z + ( 1/2 - θ ) z ^{2}}{1 - θ z} = 1 + z + \frac{z ^{2}}{2} + \frac{θ}{2} z^{3} + O (z^{4}) .

Both therefore have order two; order three would require

θ = 1/3

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