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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 17A / a / Solution

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

.

Solved by gpt-5.6-sol high.

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