 Solution

ID: past-exam-of-the-differential-equations-2-course-of-the-university-of-oxford/2017/2/a/i/solution

Use integration by parts with a complex conjugate, so that the argument also covers complex-valued eigenfunctions. The coefficient functions are understood to be real. For a nonzero eigenfunction

u

λ \int_{α}^{β} ∣ u ∣^{2} d x = \int_{α}^{β} \overline{u} M u d x = [a \overline{u} u^{'}]_{α}^{β} - \int_{α}^{β} a ∣ u^{'} ∣^{2} d x + \int_{α}^{β} b ∣ u ∣^{2} d x = - \int_{α}^{β} a ∣ u^{'} ∣^{2} d x + \int_{α}^{β} b ∣ u ∣^{2} d x .

The denominator is strictly positive and the numerator is real. Consequently the eigenvalue satisfies

λ = \frac{\int _{α}^{β} ( b ∣ u ∣ ^{2} - a ∣ u ^{'} ∣ ^{2} ) d x}{\int _{α}^{β} ∣ u ∣ ^{2} d x} \in R .

This is the Rayleigh quotient for the sign convention used here. Positivity of

a

is unnecessary for this particular reality argument; real coefficients and disappearance of the endpoint term are what matter.

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