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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