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Math 257/316, Midterm 2, Section 103 17 November 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Let f (x) be a function that has a period of 2π having the values: ½ 0 when − π ≤ x < 0 f (x) = π − x when 0 < x ≤ π (a) Determine the Fourier series expansion for f (x). (b) To what value does this series converge when x = 0? Use this value and the series in (a) to show that: π2 1 1 1 = 2 + 2 + 2 + ... 8 1 3 5 Hint: It may be useful to know that for n 6= 0: Zπ (π − x) cos(nx)dx = 1 − cos(nπ) n2 0 Zπ (π − x) sin(nx)dx = π n 0 [20 marks] 2. Solve the following inhomogeneous initial boundary value problem for the heat equation: ut = uxx + x, 0 < x < 1, t > 0 u(0, t) = 0 and u(1, t) = 0 1 u(x, 0) = sin(πx) − (x3 − x) 6 [40 marks] 3. Determine the solution to the following boundary value problem: 1 1 urr + ur + 2 uθθ r r = 0, 0 < r < a, 0 < θ < π r→0 u(r, 0) = 0 = u(r, π), u(r, θ) < ∞ u(a, θ) = sin(2θ) [40 marks] 1