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Math 257/316, Midterm 2, Section 101 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) = cos(2x) (a) Determine the sine series expansion for f (x) on the interval 0 < x < π. (b) Is the Fourier series for f (x) on the interval −π < x < π the same expansion as you found above in part (a)? Is there a way to answer this without working out the expansion? Hint: It may be useful to know that Zπ sin(nx) cos(2x)dx = 0 ½ 0 if n is even if n is odd 2n n2 −4 [20 marks] 2. Solve the following inhomogeneous initial boundary value problem for the heat equation: ut = uxx + e−t sin x, 0 < x < π, t > 0 u(0, t) = 0 and u(π, t) = 0 u(x, 0) = 0 [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 < u(r, 0) = 0 = u(r, π 2 r→0 π ), u(r, θ) < ∞ 2 u(a, θ) = sin(4θ) [40 marks] 1