 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/1/b/i/solution

Factor out

x

from the differential expression:

L y = x [- (x y^{'})^{'} - \frac{y}{x}] .

Comparing with

L y = w^{- 1} [- (p y^{'})^{'} + q y]

gives

p (x) = x, q (x) = - \frac{1}{x}, w (x) = \frac{1}{x} .

Multiplying all three by the same positive constant would give the same Sturm-Liouville operator.

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