Consider the eigenvalue problem y″ + λy = 0; y(−π) = y(π), y′(−π) = y′(π) which is not of the type in (10) because the two endpoint conditions are not “separated” between the two endpoints. (a) Show that λ0 = 0 is an eigenvalue with associated eigenfunction y0(x) = 1. (b) Show that there are no negative eigenvalues, (c) Show that the nth positive eigenvalue is n2 and that it has two linearly independent associated eigenfunctions, cos nx and sin nx.
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