 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/31a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Put

Q (x) = - q (x) = \frac{1}{4} + \frac{2}{x} + \frac{2}{x ^{2}} > 0

for sufficiently large

x

u_{\pm} (x) \sim Q (x)^{- 1/4} exp (\pm \int^{x} Q (t) d t) .

Since

Q (x) = \frac{1}{2} + \frac{2}{x} + O (x^{- 2}), Q (x)^{- 1/4} = 2 [1 + O (x^{- 1})],

we obtain, after absorbing nonzero constants into the two solutions,

u_{+} (x) \sim x^{2} e^{x /2}, u_{-} (x) \sim x^{- 2} e^{- x /2} .

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