112 年 國立中山大學電波聯合碩士班《工程數學甲》

Which of the following is true about the initial value problem of $y''' – 2y'' + y' = x e^{x} + 5$ with $y(0) = 2$, $y'(0) = 2$ and $y''(0) = -1$? Assume that the particular solution $y_p = A_1 x^{B_1} e^{C_1 x} + A_2 x^{B_2} e^{C_2 x} + A_3 x^{B_3} e^{C_3 x} + A_4 x^{B_4} e^{C_4 x}$.
(A) $\sum_{i=1}^{4}(A_i + B_i + C_i) = 41/3$.
(B) $\sum_{i=1}^{4}(A_i + B_i + C_i) = 5$.
(C) $\sum_{i=1}^{4}(A_i + B_i + C_i) = 38/3$.
(D) $\sum_{i=1}^{4}(A_i + B_i + C_i) = 14/3$.
(E) $\sum_{i=1}^{4}(A_i + B_i + C_i) = 8$.

The points A(1, -2, 1), B(0, 1, 6), and C(-3, 4, -2) form a triangle. Find the angle (°) between the line AB and the line from A to the midpoint of the line BC. (Numbers are rounded to 2 decimal places).
(A) 81.29°
(B) 121.51°
(C) 33.89°
(D) 47.40°
(E) 0.83°

Which of the following is/are linear differential equation(s)?
(A) $xy'' + (\sin x)y = 0$
(B) $y'' + 5xy' + \sqrt{x}y = 0$
(C) $\frac{d^2x}{dt^2} = t + x^2$
(D) $\frac{d^2u}{dy^2} + \frac{d^2u}{dx^2} = u$
(E) $t^2x'' - 4tx' + 6x = \ln t^2$

Consider the differential equation $(t - 4)^2x'' - 5(t - 4)x' = -9x$, which of the following is/are correct?
(A) It's an autonomous differential equation.
(B) The general solution is $x = c_1t^3 + c_2t^3 \ln t$.
(C) $t = 4$ is a regular singular point.
(D) The interval of definition of the general solution is $(0, \infty)$.
(E) The interval of definition of the general solution is $(-\infty, \infty)$.

Consider the Laplace's equation $U_{xx} + U_{yy} = 0$ with the conditions listed below. According to the boundary conditions, it's possible to reduce the Laplace's equation into two simultaneous equations with respect to x and y, separately. Assume $\lambda$ is the eigenvalue, identify the correct statements.
$\begin{cases} u(0, y) = 0, & 0 < y < H \\ u(L, y) = 0, & 0 < y < H \\ u(x, 0) = f(x), & 0 < x < L \\ u(x, H) = 0, & 0 < x < L \end{cases}$
(A) The Laplace's equation is elliptic.
(B) This question is a Dirichlet problem.
(C) The problem for X is $X'' - \lambda X = 0, X(0) = 0, X(L) = 0$.
(D) The problem for Y is $Y'' + \lambda Y = 0, Y(H) = f(0)$.
(E) The problem for Y is $Y'' + \lambda Y = 0, Y(H) = 0$.

Following question 5, which of the following is/are correct?
(A) $\lambda = \frac{n\pi}{L}$, $X(x) = \sin(\frac{n\pi x}{L})$, $n = 1,2,3, ...$
(B) $\lambda = (\frac{n\pi}{2})^2$, $X(x) = \sin(\frac{n\pi x}{2})$, $n = 1,2,3, ...$
(C) $\lambda = (\frac{n\pi}{H})^2$, $X(x) = \sin(\frac{n\pi x}{H})$, $n = 0,1,2, ...$
(D) The product solution is $\sin(\frac{n\pi x}{L}) \cosh(\frac{n\pi (y-H)}{L})$.
(E) The product solution is $\sin(\frac{n\pi x}{L}) \sinh(\frac{n\pi (y-H)}{L})$.

Which of the following is/are true about the differential equation, $y' + 2y^2 - 3y + 1 = 0$?
(A) It is a second-order homogeneous differential equation.
(B) There are two critical points.
(C) One of the critical points is an attractor.
(D) None of the critical points is an attractor.
(E) The solution y(x) increases with x when x is defined on $(1/2, \infty)$.

Consider the linear system $Ax = b$, where $A = [a_1, a_2, a_3] \in \mathbb{R}^{5 \times 3}$, $a_1, a_2, a_3$ are column vectors of A, and $b$ is a non-zero vector. Suppose $b = a_1 + a_2 = 2a_2 + a_3$. Which of the following statements are true?
(A) The linear system has exactly two solutions.
(B) The linear system has at most two solutions.
(C) $\text{rank}([A, b]) = \text{rank}(A) \leq 2$.
(D) $x = [1, 3, 1]^T$ is a solution of the linear system.
(E) The vectors $a_1, a_2, a_3$ are linearly dependent.

Consider the linear system $Ax = b$, where $A = [a_1, a_2, a_3] \in \mathbb{R}^{3 \times 3}$, $a_1, a_2, a_3$ are column vectors of A, and $b$ is a non-zero vector. Suppose $a_1 - a_2 + 3a_3 = 0$, $a_1 + 2a_2 - a_3 = b$, $\text{rank}(A) = 2$. Which of the following are solutions of the linear system?
(A) $x = [-1, 4, -7]^T$
(B) $x = [3, 0, 5]^T$
(C) $x = [2, 4, -2]^T$
(D) $x = [3, 1, 3]^T$
(E) $x = [-2, 2, -6]^T$

Two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Which of the following statements are true?
(A) If there exists a matrix R such that B = RA, then A and B are row equivalent.
(B) If two matrices are row equivalent, then they have the same null space.
(C) If two matrices are row equivalent, then they have the same eigenvalues.
(D) If E is the reduced row echelon form of A, then E and A are row equivalent.
(E) Let A ∈ $\mathbb{R}^{m \times n}$, C ∈ $\mathbb{R}^{m \times n}$, and b ∈ $\mathbb{R}^m$ be given. If A and C are row equivalent, then the two linear systems Ax = b and Cx = b have the same solution set.

Consider the linear system $Ax = b$. Let $A = QR$, where Q is an orthogonal matrix, and R is an upper triangular matrix. Suppose
$R = \begin{bmatrix} -1 & 4 & 2 \\ 0 & -3 & -1 \\ 0 & 0 & 3 \end{bmatrix}$, $Qb = \begin{bmatrix} -3 \\ 7 \\ 15 \end{bmatrix}$, $x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$.
Which of the following statements are true?
(A) $x_1 = 9$
(B) $x_2 = 1$
(C) $x_3 = 5$
(D) $x_1 + x_2 + x_3 = 12$
(E) $x_1 + x_2 + x_3 = 14$

Let $\hat{x}$ be a least-squares solution of the linear system $Ax = b$. Which of the following statements are true?
(A) $\hat{x}$ satisfies $A\hat{x} = b$.
(B) $\hat{x}$ satisfies $A^T A \hat{x} = A^T b$.
(C) $\hat{x}$ always exists.
(D) $\hat{x}$ is unique.
(E) $b - A\hat{x}$ is orthogonal to any vector in the range space of A.

Let $x_1$ and $x_2$ be eigenvectors of A, and $\lambda_1$ and $\lambda_2$ be the corresponding eigenvalues. Suppose
$x_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$, $x_2 = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$, $\lambda_1 = 2$, $\lambda_2 = -1$, $y = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$, $A^{10}y = \begin{bmatrix} S_1 \\ S_2 \end{bmatrix}$.
Which of the following statements are true?
(A) $S_1 - 2045$
(B) $S_1 - 2046$
(C) $S_2 = 4096$
(D) $S_2 = 4098$
(E) $S_1 + S_2 = 2052$

(10%) Use the Laplace transform to solve the following initial value problem.
$y'' + 6y' + 9y = e^{-3t} \cos t$, $y(0) = 0$, $y'(0) = 0$

(10%) The Fourier transform of a continuous-time real function is given as
$Y(j\omega) = \int_{-\infty}^{\infty} y(t) e^{-j\omega t} dt$.
(a) Obtain the Fourier transform of the following function. (5%)
$y(t) = \left(\frac{\sin 100\pi t}{\pi t}\right)^2$
(b) To sample y(t) with equal interval, what is the maximal sampling period T to possibly prevent from aliasing? (5%)

Let $C$ be the circle $|z^*+j|=2$ oriented in a positive sense, where $z^*$ indicates the complex conjugate of $z$, and a complex integral $g$ is defined as
$g(z_0)=\int_C\frac{z^2-z_0}{z^2+z_0}\,dz.$