105年 - 105 國立臺北大學_日間學士班轉學生招生考試試題_電機工程學系日間學士班3年級：工程數學#147416

(a) \( (5\%) \) \( \frac{dy}{dt} = -0.5y, y(0) = 3 \)

(b) \( (5\%) \) \( \frac{dy}{dt} = 50 - 0.01y, y(0) = 100 \)

(c) \( (5\%) \) \( \frac{dy}{dx} = (x + y - 2)^2, y(0) = 2 \)

(a) \( (5\%) \) \( y^{\prime \prime} + y = 0,y(0) = 3,y^{\prime}(0) = -0.5 \)

(b) \( (5\%) \) \( y^{\prime \prime} + y^{\prime} + 0.25y = 0,y(0) = 3,y^{\prime}(0) = -3.5 \)

(c) \( (5\%) \) \( x^{2}y^{\prime \prime} + xy^{\prime} + 9y = 0,y(1) = 0,y^{\prime}(1) = 2.5 \)

3. (15%) Solve the initial value problem by using the Laplace transform.
y'' + 4y' + 3y = f(x), y(0) = y'(0) = 0,
where f(x) = 5x, if 0 < x < 2 and f(x) = 10, if 2 < x.

4. (10%) Find the directional derivative of \( f(x, y, z) = 2x^2 + 3y^2 + z^2 \) at point \( P \): (2, 1, 3) in the direction of \( a = [1, 0, -2] \).

(a) \( (5\%) \) Show that the force is conservative.

(b) \( (5\%) \) Find the potential function \( \Phi (x,y,z) \) for this force field.

(c) (5%) Calculate the work done by this field upon a particle moving from position \( (1, -2, 1) \) to position \( (3, 1, 4) \).

6. (15%) Find the Fourier series of the following periodic function f(x) with period $2\pi$. $$f(x) = \begin{cases} -1, & -\pi < x < 0 \\ 1, & 0 < x < \pi \end{cases}$$

7. (15%) Integrate counterclockwise
$\oint_C (z + z^{-1}) dz,$
where $C$ is the unit circle.