Consider the eigenvalue problem
y″ + λy = 0; y (0) = 0, y (1) = y′ (1);
all its eigenvalues are nonnegative. (a) Show that λ = 0 is an eigenvalue with associated eigenfunction y0(x) = x. (b) Show that the remaining eigenfunctions are given by yn(x) = sinβnx, where βn is the nth positive root of the equation tanz = z. Draw a sketch showing these roots. Deduce from this sketch that βn ~ (2n + 1 )π/2 when n is large.
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