107年 - 107 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#147720

(a) Find the inverse Laplace transform of \( \frac{s}{(s^2 + 4)(s + 2)} \). (5%)

(b) Find the Laplace transform of \( (t - 3)^2 u(t - 4) \). (5%)

2. Solve \( y^{*} - 6y' + 9y = 6x^{2} + 2 - 12e^{3x} \). (10%)

3. Solve \( (xy)dx + (2x^{2} + 3y^{2} - 20)dy = 0 \). (10%)

4. Solve \( (x - 2)^{2}y^{*} - 5(x - 2)y' + 8y = 0 \), \( y(4) = 32 \), \( y'(4) = 0 \). (10%)

(a) The half-wave rectification of \( \sin (\pi t) \) given in Fig. 1(a). (5%)

(b) The full-wave rectification of \( \sin (\pi t) \) given in Fig. 1(b). (5%)

(a) in a Fourier cosine series. (5%)

(b) in a Fourier sine series. (5%)

7. Calculate the current \( i(t) \) of series RL circuit shown in Fig. 3 if the initial current \( i(0) = 1\mathrm{A} \) and the impressed voltage is \( v_{i}(t) = V_{0}[u(t - a) - u(t - b)] \). (Assume \( V_{0} = 5\mathrm{V}, R = 2\Omega \), and \( L = 0.02\mathrm{H}, a = 2\mathrm{sec}, b = 4\mathrm{sec} \)). Roughly draw the waveform \( i(t) \). (10%)

8. The following system of equations has a unique solution. Solve this system using the method of Gauss-Jordan elimination with matrix. (10%)
\( x_1 - x_2 + 3x_3 = 3 \\ 2x_1 - x_2 + 2x_3 = 2 \\ 3x_1 + x_2 - 2x_3 = 3 \)

9. Solve the following system of three equations in three variables by determining the inverse of the matrix of coefficients and then using matrix multiplication. (10%)
\( x_1 + 2x_2 + 3x_3 = 1 \\ 2x_1 + 5x_2 + 3x_3 = 3 \\ x_1 + 8x_3 = 15 \)

10. Determine the eigenvalues of the given \( 3 \times 3 \) matrix. (10%)
\( \begin{bmatrix} 3 & 2 & -2 \\ -3 & -1 & 3 \\ 1 & 2 & 0 \end{bmatrix} \)