final - MATH 152, FALL 2004: FINAL There are five problems....

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MATH 152, FALL 2004: FINAL There are five problems. Do all of them. Total score: 160 points. Problem #1, (25 points) For both of the following functions f on [0, 11, state whether the Fourier cosine series on [0, I] converges in each of the following senses: uniformly, pointwise, in L2. If the Fourier series converges pointwise, state what it converges to for each x E [0, I]. Make sure that you give the reasoning that led you to the conclusions. 1. f(x) = x(sin(n-X/L))~, 2. f (x) = 0, for 0 5 x 5 112, and f (x) = 1 for 112 < x 5 1. Problem #2, (25 points total) 1. (8 points) Find the general solution of the PDE u, + 2yuy = 0 on R, x Ry. 2. (7points) Now impose in addition that u(0, y) = y. Find u explicitly. 3. (10 points) Consider the PDE Urn + 2yuy = x on IR, x R,. Find its general solution. Problem #3, (35 points total) Consider the differential operator on twice differentiable functions f on [0, I] which satisfy Dirichlet boundary con- ditions f (0) = f (1) = 0. That is, for these functions f, A f = -(x2 fl(x))'. Let 1 - (f, g) = So f (x)g(x) dx denote the standard inner product on functions. 1. (10 points)
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final - MATH 152, FALL 2004: FINAL There are five problems....

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