(a) Consider the following eigenvalue problem: X" + XX=0 X'(0) = 0, X (π) = 0. (i) Show that all the eigenvalues satisfy >> 0. (ii) Find all eigenvalues and eigenfunctions. + (b) Solve the following wave equation with mixed boundary conditions on an interval. (You can make use of the results obtained in (a).) Utt - c²Uxx = 0 Ur(0, t) = 0, U(π, t) = 0 (U(x, 0) = 0, Ut(x, 0) = 6c. cos (3x).

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