109年 - 109 國立嘉義大學_碩士班招生考試試題_電機工程學系：工程數學#146116

(a) Find the inverse Laplace transform of\[ \frac{2s-1}{(s-1)(s^2+4s+3)} \]
. (10%)

(b) Find the Laplace transform of \( f(t) = (t-2)^2 u(t-3) \), where \( u(t) \) is unit step function. (10%)

2. Solve \( 6xydx + (4y + 9x^2)dy = 0 \). (20%)

3. Expand the periodic function f(x) shown in Fig. 1 in a Fourier trigonometric series. (20%)

4. The following system of equations has unique solution. Solve this system using the method of Gauss-Jordan elimination with matrices. (20%)

(a) \[ \begin{bmatrix} 1 & 3 \\ 3 & 9 \end{bmatrix} \]

(b) We know that if \( D = C^{-1}AC \), then \( A^k = CD^kC^{-1} \). Use this result to compute the following power. \[ \begin{bmatrix} 1 & 3 \\ 3 & 9 \end{bmatrix}^6 \] , (10%)