 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/31e/iv/solution

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

Since

y = x^{- 1/2} u

, the two linearly independent solutions of the original equation have the asymptotic expansions

y_{grow} (x) y_{decay} (x) \sim x^{1/2} e^{1/ x} (1 + \frac{x}{8} + \frac{9 x ^{2}}{128} + \dots), \sim x^{1/2} e^{- 1/ x} (1 - \frac{x}{8} + \frac{9 x ^{2}}{128} + \dots) .

As a check, the substitution

z = 1/ x

converts the equation into the modified Bessel equation of order zero, whose solutions

I_{0} (z)

and

K_{0} (z)

have exactly these large-

z

expansions.

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