Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 17B / b

Solution

With

q = p^{'}

and

r = 0

, the equation is

y^{''''} + (p y^{'})^{'} = λ y .

Multiply by the real eigenfunction

y

and integrate over

[a, b]

y = y^{'} = 0

at both endpoints eliminate all boundary terms, so

λ \int_{a}^{b} y^{2} d x = \int_{a}^{b} (y^{''})^{2} d x - \int_{a}^{b} p (x) (y^{'})^{2} d x .

Since

p (x) < 0

, the right side is a sum of nonnegative terms. It cannot vanish for a nonzero eigenfunction: if

y^{''} = y^{'} = 0

y = 0

. Thus

λ = \frac{\int _{a}^{b} {( y ^{''} ) ^{2} - p ( y ^{'} ) ^{2} } d x}{\int _{a}^{b} y ^{2} d x} > 0 .

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