Consider the eigenvalue problemy″ + λy = 0; y′(0) = 0, y (1) + y′(1) = 0.All the eigenvalu...

Consider the eigenvalue problem

y″ + λy = 0; y′(0) = 0, y (1) + y′(1) = 0.

All the eigenvalues are nonnegative, so write λ = α2 where α ≥ 0. (a) Show that λ = 0 is not an eigenvalue. (b) Show that y = A cos αx + B sin αx satisfies the endpoint conditions if and only if B = 0 and α is a positive root of the equation tan z = 1/z. These roots are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. Thus the eigenvalues and eigenfunctions of this problem are the numbers {α2n}∞1 and the functions {cos αn x}1∞, respectively.