Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 1 / b

Solution

y

and integrate. Integration by parts gives

λ \int_{a}^{b} ω y^{2} d x = \int_{a}^{b} y L y d x = - [p y y^{'}]_{a}^{b} + \int_{a}^{b} [p (y^{'})^{2} + q y^{2}] d x .

Since

y

is a nonzero eigenfunction and

ω > 0

, the denominator is strictly positive. Thus the Rayleigh quotient is

λ = \frac{\int _{a}^{b} [ p ( y ^{'} ) ^{2} + q y ^{2} ] d x - [ p y y ^{'} ] _{a}^{b}}{\int _{a}^{b} ω y ^{2} d x} .

The endpoint term must be retained for a general Robin boundary condition; it vanishes for homogeneous Dirichlet boundary conditions but need not vanish in the present general problem.

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