課程名稱︰工程數學下 課程性質︰必修 課程教師︰施文彬 開課學院：工學院 開課系所︰機械工程學系 考試日期（年月日）︰104/5/18 考試時限（分鐘）：110分鐘 試題 : Prelim II, Engineering Mathematics II, Spring 2015 Time: 10:20 ~ 12:10 noon, May 18, 2015. Rule: No calculators are allowed. Points will not be given without providing details of your calculation. Please carry out all integrations in your calculation. Good luck! l. (20%) Solve y_tt = y_xx - sin(3x) for 0 < x < 2π, for t > 0; y(0,t) = y(2π,t) = 0 for t 0; y(x,0) = 0, y_t(x,0) = x for 0 < x < 2π. 2. f has period 3 and f(x) = cosh (x) for 0 ≦ x < 3. (a)(8%) Determine the phase angle form of Fourier series of f. (b)(4%) Determine what this series converges to for -4 ≦ x ≦ 1. (d)(3%) Plot some points of the amplitude spectrum. 2a 3. (a)(10%) Given F[e^(-a|t|)] = ────── use Fourier transform to solve -y" + y = e^(-2|t|). a^2 + ω^2 ∞ (a)(10%) Evaluate ∫ δ(t-3)H(t-3)e^(-5t)dt -∞ ^ 4. (a)(5%) Given F[f(t)] = f(ω), prove that i ^ ^ F[f(t)sin(ω_0t)] = ---[f(ω+ω_0) - f(ω-ω_0)] 2 (b)(7%) Following (a) and given F[δ(t)] = 1, determine F[sin(2t)](ω). -at 1 (c)(8%) Following (b) and given F[H(t)e ] = ────, determine a+iω -1 sin(2ω) F [─────](t). 2 + iω 5. (a)(15%) solve (x^(-3)y')' + (λ + 4)x^(-5)y = 0; y(1) = y(e^2) = 0. (b)(10%) Find the first two non-zero coefficients in the Fourier-Legendre expansion of the function on [-1,1]: f(x) = cos(πx/2). http://imgur.com/oLeDq39 --

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