课程名称︰工程数学下
课程性质︰必修
课程教师︰施文彬
开课学院：工学院
开课系所︰机械工程学系
考试日期（年月日）︰104/5/18
考试时限（分钟）：110分钟
试题 :
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)] =