 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/8/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Write the analytic solution as

w (z) = \sum_{n \geq 0} a_{n} z^{n}

a_{1} = 0

, and matching the coefficient of

z^{n + 1}

gives

(n + 2)^{2} a_{n + 2} + 4 a_{n} = 0, n \geq 0.

All odd coefficients therefore vanish;

a_{2} = - a_{0}

and

a_{4} = a_{0} /4

. The first three nonzero terms are

w (z) = a_{0} (1 - z^{2} + \frac{z ^{4}}{4} + O (z^{6})) .

In full,

a_{2 j} = (- 1)^{j} a_{0} / (j!)^{2}

, so

w = a_{0} J_{0} (2 z)

, a Bessel function of the first kind. The ratio test proves convergence for every complex

z

. The other independent solution has a logarithmic singularity at zero and does not satisfy the analyticity requirement.

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