104年 - 104 國立臺北大學_日間學士班曁進修學士班轉學生招生考試試題_通訊工程學系日間學士班3年級：工程數學#147423

(1) \( z = \overline{z_1} / \overline{z_2} \), Let \( z_1 = -2 - 10i \), \( z_2 = 5 - i \)

(2) \( z = \left(\overline{z_1 / z_2}\right) \), Let \( z_1 = -2 - 10i \), \( z_2 = 5 - i \)

(3) \( \ln z = e + \pi i \)

(4) \( \ln z = 0.4 + 0.2i \)

(1) \( f(z) = e^{z}(\cos y + i\sin y) \)

(2) \( f(z) = z^2 \)

(1) \( (12\%) \) find two linearly independent solutions and compute their Wronskian;

(2) \( (6\%) \) find the solution if \( y^{\prime}(0) = y^{\prime \prime}(0) = 4 \)

4. (10%) Solve the following differential equation $$y''(x) - 4y'(x) + 4y(x) = \frac{e^{2x}}{x^4}.$$

(1) \( (6\%) \) find the Laplace transform of \( r(t) \)

(2) \( (6\%) \) solve the differential equation if \( y(0) = y'(0) = 0 \).

6. (10%) The function $f(x)$ be partially defined by $$f(x) = \begin{cases} \frac{\pi}{2}, & 0 < x \le \frac{\pi}{2}, \\ \pi - x, & \frac{\pi}{2} < x < \pi. \end{cases}$$ If $f(x)$ is an odd periodic function with period $2\pi$, find its Fourier series representation.

7. (10%) Given the vector function $f(x, y, z) = [x^2 + y^2, 3xy + z^2, 4z^2]$, find $\operatorname{div} f$ and $\operatorname{curl} f$.