Consider u(x, I) that solves the heat or wave equation and is periodic in x with period 2L. This could describe a thin circle of wire with length 2L, for instance. Use the spatial coordinate x E [-L, L]. Since x = L and x = L are really the same point, all values of u are the same there, as are any well-defined derivatives of 1.4. Assume that at least ax and um are continuous, which is necessary to have a classical solution of the wave or heat equation. (3) Suppose you look for separable solutions u(x, t) = T(t)X (x) to the wave or heat equation, which requires X \" (x) = 11X (x) for some 1 E R. Show that if a single eigenvalue :1 has two corresponding eigenfunctions X1 (x) and X 2 (x), then any linear combination of X1 and X2 is also an eigenfunction. (If X1 and X2 are linearly independent, then we say that A has multiplicity 2.} (b) The eigenvalue 110 = 0 has multiplicity 1; all of its eigenfunctions are proportional to X0 (x) = 1. What are the other eigenvalues 21,, for n = l, 2, .. . ? (0) Let n = 1/2,, for each n 2 1. Show that sin(,,x - y) is an eigenfunction corresponding to 11,, for any 1/ E IR. (d) Show that sin(nx - y) is in the span of sin(,,x) and cos(nx). (9) Show that sin(,,x y) with 0