MATH 257 Study Guide - Midterm Guide: The Vibrations, Regular Singular Point, Fourier Series
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Do not open this test until instructed to do so! This exam should have 19 pages, including this cover sheet. No textbooks, calculators, or other aids are allowed. Turn o any cell phones, pagers, etc. that could make noise during the exam. Use the back of the page if necessary. Problem 1 (total 15 points) let f (x) =( 0. Consider the di erential equation x2y + xy +(cid:18)x2 . Find the solution u(x, t) of the following heat equation ut = 9uxx, u(0, t) = 2, u(x, 0) = 2x. 0 < x < 1, t > 0 u(1, t) = 0 (a) [5] find the steady-state solution v(x). (b) [15] find the solution u(x, t). Find the solution u(r, ) of the following laplace equation in a pie-shaped domain: ur + urr + 3 u(r, ) bounded for r 0, ur(1, ) = f ( ) =( 1 if 0 < < .
