104年 - 104 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_電機工程學系、通訊工程學系學士班3年級：工程數學#147392

1. (8%) Solve the following ODE $$(x^2 + 3y^2)dx - 4xydy = 0$$ where y is a function of x.

2. (12%) Let $y = (A + Bx)e^{ax} + Ce^{2x}$ be the general solution to the following linear constant-coefficient ODE $$y''' + py' + qy = 0.$$ Solve a, p and q. 

3. (12%) (a) Show that $y = e^x$ is a solution to the following ODE $$(x - 1)y'' - xy' + y = 0.$$

(b) Find a basis of solutions of the above ODE.

4. (8%) Let f(x) be a periodic function with period $2\pi$ and defined by $$f(x) = \begin{cases} x + \pi, & |x| \le \frac{\pi}{2}, \\ 0, & \frac{\pi}{2} < |x| \le \pi. \end{cases}$$ for $-\pi \le x \le \pi$. Find the Fourier series of f(x).

5. (10%) Solve the following initial value problem by using the Laplace transform. $$y_1'(x) + y_2(x) = x, \quad y_1(x) + y_2'(x) = 0,$$ $$y_1(0) = 1, \quad y_2(0) = 0$$

(a) \( (5\%) \) What are Cauchy-Riemann Equations?

(b) \( (5\%) \) What is the purpose of Cauchy-Riemann Equations?

(a) \( (5\%) f(z) = e^{-x}\cos y - i e^{-x}\sin y \)

(b) \( (5\%) f(z) = \operatorname{Arg} z \)

8. (10%) Integrate counterclockwise. Show the details.
$\oint_C \frac{z}{z^2 + 4z + 3} dz$, C the circle with center -1 and radius 2.

9. (10%) Integrate counterclockwise. Show the details.
$\oint_C \frac{e^x}{2e^x - 2iz} dz, C: |z| = 0.6$

10. (10%) Find the principal value
$\text{pr. v. } \int_{-\infty}^{\infty} \frac{dx}{(x^2 - 3x + 2)(x^2 + 1)}.$