Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 14A / b / i

Table of contents
- Solution i

Solution

Multiply the equation by

x^{2}

and combine the derivative terms:

(x^{3} y^{'})^{'} + (1 + λ) x y = 0.

Thus the self-adjoint form is

- (x^{3} y^{'})^{'} - x y = λ x y, p = x^{3}, q = - x, w = x .

The Dirichlet boundary conditions remain at

1

and

e

, and the weight

w = x

is positive on this interval, as required for the Sturm-Liouville problem.

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