 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-numerical-solution-of-differential-equations-course-of-the-university-of-oxford/2020/1/a/solution

Φ (x, v, h) = f (x + β h, v + β h f (x, v))

T_{n} = \frac{y ( x _{n} + h ) - y ( x _{n} )}{h} - Φ (x_{n}, y (x_{n}), h) .

h \to 0

, the first term tends to

y^{'} (x_{n})

and the second tends to

f (x_{n}, y (x_{n})) = y^{'} (x_{n})

by continuity. Both limits are uniform along the exact-solution curve, a compact set: its small stage perturbations lie in a fixed compact neighborhood, so uniform continuity applies there. Thus

max_{n < N} ∣ T_{n} ∣ \to 0

, proving consistency for every $β \in [0, 1]$ .

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