110年 - 110 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#146536

1. y”+4y’+3y=et ; y(0)=0, y’(0)=2 (10%)

2.To inverse the Fourier transformation of \( \displaystyle \frac{1+\omega i}{6-\omega^2+5\omega i} \)(10%)

3. To obtain \( f(t) \) when \( f(t)=2t^{2}+\int_{0}^{t} f(t-\alpha) e^{-\alpha} d\alpha \) (20%)

4. To obtain the Laplace transform of f(t) as shown below (20%)

5. To obtain the Fourier transformation of \( f(t) = e^{-a|t|} \) (10%)

6. The following system of equations has unique solution. Solve the system using the method of Gauss-Jordan elimination with matrix. (10%) \[ \begin{aligned} x_1 - x_2 + 3x_3 &= 3 \\ 2x_1 - x_2 + 2x_3 &= 2 \\ 3x_1 + x_2 - 2x_3 &= 3 \end{aligned} \]

7. Determine the characteristic polynomial, eigenvalues, and corresponding eigenspaces of the following matrix. (10%) \[ \begin{bmatrix} 5 & -7 & 7 \\ 4 & -3 & 4 \\ 4 & -1 & 2 \end{bmatrix} \]

8. We know that if \( D = C^{-1}AC \), then \( A^k = CD^kC^{-1} \). Use this result to compute the following power. (10%) \[ \begin{bmatrix} 1.5 & -0.5 \\ -0.5 & 1.5 \end{bmatrix}^{16} \]