[試題] 108-1 顏炳郎 工程數學 期末考
課程名稱︰工程數學
課程性質︰生機系必修
課程教師︰顏炳郎
開課學院：生農學院
開課系所︰生機系
考試日期（年月日）︰109.01.07
考試時限（分鐘）：180
試題 :
1.(10%) Use the Gram-Schmidt process to find an orthogonal basis from the fol-
┌1┐┌1┐┌2┐
lowing linearly independent vectors: {│0││1││3│}
└0┘└0┘└2┘
┌1 -2 1 2┐ ┌-11┐
2.(10%) Solve Ax = b, where A = │0 1 -1 -1│, b = │ 5│, and also find
│1 1 1 0│ │ 4│
└3 2 0 1┘ └ 7┘
A^-1 = ?
3.(10%) Solve the following DE: X' = ┌-5 9┐X + ┌e^t ┐
└-6 1┘ └e^{-2t}┘
┌-3 1┐
4.(10%) Find e^{└ 2 -4┘t}
5.(10%) Find the general solution of the non-homogeneous system using variati-
on of parameter.
┌1 1 2 0┐ ┌ e^t┐
X' = │0 1 3 0│X + │ t│
│0 0 2 2│ │te^t│
└0 0 0 1┘ └ t^2┘
6.(10%) y'' + 8y' + 16y = t^2e^{-4t}; y(0) = 1, y'(0) = -4
7.(10%) Find the Laplace transform of the following functions in time domain:
t
f(t) = ∫e^{t-τ}(t-1)^{3/2}dτ
0
8.(10%) Find the inverse Laplace transform of the following function in s-dom-
ain:
F(s) = (s^2e^{-πs/2}) / (s+1)(s^2+4)
9.(10%) Solve an initial value problem: y' + 2y = f(t), y(0) = 1, where the
input function f(t) is a periodic function defined as:
f(t) = 2 for 2n ≦ t ＜ 2n+1 , where n is a non-negative integer
0 for (2n+1) ≦ t ＜ 2(n+1)
10.(10%) Solve the following initial value problem:
y'' + 6y' + 5y = t - tu(t-2), y(0) = 1, y'(0) = 0, where u(t) is unit step fu-
nction. Please identify the zero input response and zero state response.
Bonus:
1.(10%) Find the K to minimize P - KHP -PH^TK^T + KSK^T
2.(10%) Using Laplace Transform to solve the initial value problem:
tx'' + x' + tx = 0, x(0) = 1, x'(0) = 0
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