 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/31a/c/solution

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

u_{\pm} \sim q^{- 1/4} exp (\pm i \int^{x} q d s) .

Here

q^{- 1/4} = x^{1/4} (1 + 1/ (4 x) + 5/ (32 x^{2}) + \dots),

\int^{x} q d s = 2 x + x^{- 1/2} + x^{- 3/2} /12 + \dots .

The control ratio

∣ q'∣ q^{- 3/2} = O (x^{- 1/2})

tends to zero, verifying consistency.

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