102年 - 102 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#146375

1. Solve \((x - y³ + y² \sin x)dx - (3xy² + 2y \cos x)dy = 0\). (10%)

2. Solve \( 4y'' - 4y' - 3y = \cos 2x \). (15%)

(a) Find the Laplace transform of the function \( (t^3 - 3t + 2)e^{-2t} \). (8%)

(b) Find the inverse Laplace transform of \( \frac{12}{s^2 + 4s + 20} \). (10%)

(a) Write down the governing equation for the current i(t).　　 (5%)

(b) Find \( i(t) \) if \( R = 2 \ \Omega \), \( L = 1 \ \text{H} \), \( C = 0.5 \ \text{F} \), and \( v_s(t) = \begin{cases} 1 \ \text{kV}, & 0 < t < 2 \\ 0, & 2 < t \end{cases} \). (10%)

5. Find the Fourier series of the given function \( f(x) \), which is assumed to have the period \( 2\pi \), where \( f(x) = x + \pi \) for \( -\pi < x < \pi \). (10%)

(a) Find the rank, a basis for the row space and a basis for the column space of A. (10%)

(b) Find all the eigenvalues and their corresponding eigenvectors of B. (10%)

7. Solve the heat equation \( k \frac{\partial^2 u(x,t)}{\partial x^2} = \frac{\partial u(x,t)}{\partial t} \), \( 0 < x < L \), \( t > 0 \) subject to the given conditions \( u(0,t) = 0 \), \( u(L,t) = 0 \), and \( u(x,0) = \begin{cases} 1, & 0 < x < L/2 \\ 0, & L/2 < x < L \end{cases} \). (12%)