 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-8/2/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

w_{i} (x)

and then the exponential eigenfunction convention. For every integer

n

\partial^{n} (w_{i} (x) e^{k x}) = e^{k x} \sum_{r \geq 0} (r n) w_{i}^{(r)} (x) k^{n - r} .

Consequently

Pw = k^{β} e^{k x} \sum_{j, i, r \geq 0} g_{j} (r α - j) w_{i}^{(r)} k^{α - j - i - r} .

Collecting all terms of a fixed power of the spectral parameter gives the useful answer

Pw = k^{α + β} e^{k x} s \geq 0 \sum \frac{v _{s} ( x )}{k ^{s}}, v_{s} = i + j + r = s \sum g_{j} (r α - j) w_{i}^{(r)} .

The sum defining each

v_{s}

is finite. For example,

v_{0} = g_{0} w_{0}, v_{1} = g_{0} w_{1} + g_{1} w_{0} + α g_{0} w_{0}^{'},

and

v_{2} = g_{0} w_{2} + g_{1} w_{1} + g_{2} w_{0} + α g_{0} w_{1}^{'} + (α - 1) g_{1} w_{0}^{'} + (2 α) g_{0} w_{0}^{''} .

k = \infty

, with

k^{β}

carried as a formal prefactor. It does not assert convergence for finite

k

, and negative powers are not evaluated at

k = 0

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