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Name:Math 54, Summer 2009, Lecture 4“Quiz 13”(1) Consider the PDE uxx+ ux− ut= 0. Derive a pair of ODEs that X(x) and T (t) wouldhave to satisfy for u(x, t) = X(x)T (t) to satisfy this PDE. (3 points)1(2) (a) Let f(x) = (1 − x)(ex− 1). Set up the Fourier series for f on [−π, π], and the Fouriersine and cosine series for f on [0, π]. By “set up”, I mean that you do not need to evaluateany integrals, just write them down. (4 points)(b) Write down a formal solution to the heat problemut= uxx0 < x < π, t > 0,u(0, t) = u(π, t) = 0 t > 0,u(x, 0) = f(x) 0 < x < π,where f (x) is as in (a). Again, do not evaluate any of the integrals. (2
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