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)] =