108年 - 108 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#146537

(a) Find the inverse Laplace transform of\[ \frac{5}{s(s^2 + 36)} \] . (5%)

(b) Find the Laplace transform of f(t) = t + (1−4t)u(t−3), where u(t) is unit step function. (5%)

2. y′′ + 4 y =
8 x 2 , y (0) = −3, y′(0) = 0 (10%)
2 x+ y

3. Solve \( (2e^{2x+y} + 3ye^y)dx + (3xe^y - 4)dy = 0 \quad (10\%) \)

4. Expand the periodic function f(x) shown in Fig. 1 in a Fourier trigonometric series. (10%)

5. Solve \( y(t) - \int_{0}^{t} y(\tau)(t - \tau) d\tau = 2 - \frac{1}{2}t^{2} \) (10%)

6.Find the Fourier transform of \( f(x) = xe^{-x^2} \) (10%)

7. Solve \( y'' + 2y' - 24y = 8 - (x + 2)e^{4x} \) (10%)

8.The following system of equations has unique solutions. Solve this system using the method of Gauss-Jordan elimination with matrices. (10%) \[ \begin{aligned} x - 2y + 3z &= 9 \\ -x + 3y &= -4 \\ x - 5y + 5z &= 17 \end{aligned} \]

9. Use the matrix inverse method to solve the following system of equations. (10%)
2x + 3 y + z = −1
3x + 3 y + z = 1
2x + 4y + z = −2

10. Find the determinant of A below (10%) \[ \mathbf{A}= \begin{bmatrix} 1 & -2 & 3 & 0 \\ -1 & 1 & 0 & 2 \\ 0 & 2 & 0 & 3 \\ 3 & 4 & 0 & -2 \end{bmatrix} \]