Consider the eigenvalue problem
y″ + λy = 0; y(0) = 0, y(1) + y′(1) = 0;
all its eigenvalues are nonnegative. (a) Show that λ = 0 is not an eigenvalue. (b) Show that the eigenfunctions are the functions {sinαnx}∞1, where αn is the nth positive root of the equation tan z = −z. (c) Draw a sketch indicating the roots {αn}∞1 as the points of intersection of the curves y = tanz and y = −z. Deduce from this sketch that αn~(2n-1)π/2 when n is large.
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