[試題] 103下 施文彬 工程數學下 第二次期中考

[NTUkobe (台大科比)]

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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※ 編輯: NTUkobe (140.112.73.180), 05/21/2015 00:34:58
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