Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 326 / 2 / b

Table of contents
- Solution b

Solution

Suppose

K f = λ f

. Differentiating the integral expression on the two sides of

y = x

gives

(K f)^{''} = K f - 2 f, (K f)^{'} (0) = K f (0), (K f)^{'} (1) = - K f (1) .

Therefore

f^{''} = \frac{λ - 2}{λ} f = - ω^{2} f, f^{'} (0) = f (0), f^{'} (1) = - f (1) .

The first boundary condition makes

f (x) = A (cos (ω x) + \frac{1}{ω} sin (ω x)) .

Substitution into the second gives

2 cos ω + (\frac{1}{ω} - ω) sin ω = 0,

or

\frac{2 ω}{ω ^{2} - 1} = tan ω, - ω^{2} = \frac{λ - 2}{λ} .

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