107年 - 107 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、資訊工程學系日間學士班3年級：工程數學#147518

1. (10%) Solve the ODE \( xy' - y = x^3 e^{y / x} \).

(a) (10%) Find another linearly independent solution \( y_{2}(x) \).

(b) (5%) Compute the Wronskian of \( y_{1}(x) \) and \( y_{2}(x) \).

3. (10%) Solve the ODE \( y'' + 9y = 6\sin(3x), y(0) = 1, y'(0) = 2 \).

4. (10%) Solve the ODE \( y'' + 4y' - 21y = r(t) \) where \( r(t) = 1 \) for \( 0 < t \leq 2 \) and \( r(t) = 0 \) for \( t > 2 \) or \( t < 0 \), and the initial condition \( y(0) = 0, y'(0) = 0 \).

5. (15%) Let \( f(x) \) be an odd periodic function with period \( 2\pi \). \( f(x) = 0 \) for \( 0 < x < \pi / 2 \) and \( f(x) = x \) for \( \pi / 2 < x < \pi \). Find the Fourier series representation of \( f(x) \).

(a)(4%) \( \pmb{a}\cdot \pmb{b} \)

(b)(4%) \( \pmb{b} \times \pmb{c} \)

(c)(4%) \( \mathbf{c}\cdot (\pmb {b}\times \pmb{a}) \)

7. (8%) Let \( f(x, y, z) = \sqrt{x^2 + 2y^2 + 3z^2} \). Find \( \nabla f \) and the value of \( \nabla f \) at the point \( (1, -2, 3) \).

(a) \( (4\%) \) div \( \pmb{g} \)

(b) \( (4\%) \) curl \( \pmb{g} \)

9. (12%) Let f(x) be an even function. f(x) = 2 for $0 \leq x < 3$ and f(x) = 0 for x > 3. Find the Fourier integral representation of f(x).