课程名称︰工程数学一 课程性质︰必修 课程教师︰徐治平 开课学院：工学院 开课系所︰化工系 考试日期（年月日）︰101/1/13 考试时限（分钟）：10:20 ~ 12:10 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.Find the Laplace transform of f(t) (6% each) t 1 - cosh(au) a)∫ ──────── du 0 u b)(t^2)sin(ωt) c)f(t)= ╭ 1 , 0＜t＜1 │ 0 , 1＜t＜2π ╰ cost , 2π＜t d) f(t) │ │ │ │ 2│ ∕╲ │ ∕ ╲ │ ∕ ╲ │ ∕ ╲ │ ∕ ╲ │ ∕ ╲ └─────────────── t 0 1 2 4 2.Find the inverse Laplace transform of F(s) = Lf(t) (8% each) 6s-4 a)F(s)= ────── s^2-4s-20 (s-1)exp(-s) b)F(s)= ─────── s(s+1) s c)F(s)= ────── (s^2-1)^2 12s-24 d)F(s)= ──────── (hint:use convolution theorem) (s^2+4s+40)^2 1 e)F(s)= ──────── (hint:use convolution theorem) (s^2+4s+13)^2 3.Using the method of Laplace Transformation to solve initial value problem: (12% each) d^2 y d^2 z dy │ a) ──── - 3 ──── + 2y = 4t with y(0)=1 , ──│ = -1 d t^2 d t^2 dt │t=0 b) ╭ d^2 y d^2 z dz │ ──── - ──── + ── - y = f_1(t) │ d t^2 d t^2 dt │ │ d^2 y d^2 z dy │ 2 ──── - ──── - 2 ── + z = f_2(t) │ d t^2 d t^2 dt ╰ dy dz ╭ f_1(t) = exp(t) - 2 with y(0) = ── = z(0) = ── = 0 , where │ dt dt ╰ f_2(t) = -t 4.Use Laplace Transformation to solve the integral equation: (12%) t y(t)= t - 2 + cost + y'(t) + ∫ cos(t-u)y(u)du 0 --

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