 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/1/ii/a/solution

x

and collect the first two derivative terms into a single derivative:

- (x y^{'})^{'} - \frac{y}{x} = λ \frac{y}{x} .

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

Indeed

w^{- 1} [- (p y^{'})^{'} + q y] = - x^{2} y^{''} - x y^{'} - y

. Both

p

and

w

are positive and regular throughout this interval. The zero endpoint values make the boundary form vanish.

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