109 年 國立臺灣大學醫療器材與醫學影像研究所《工程數學（C）》

Which are the possible solutions of the differential equation
$\frac{dy}{dx} = y^2 - 9$
(A) $y = 3$
(B) $y = -3$
(C) $y = 3 \frac{1+e^{6x}}{1-e^{6x}}$
(D) $y = 3 \frac{1+e^{-6x}}{1-e^{-6x}}$
(E) $y = 3 \frac{1-e^{-6x}}{1+e^{-6x}}$

Which are the possible solutions of the differential equation
$\frac{dy}{dx} + y = e^{-2x}$
(A) $y = e^x$
(B) $y = e^{-2x}$
(C) $y = e^x - e^{-2x}$
(D) $y = -e^x - e^{-2x}$
(E) $y = e^x + e^{-2x}$

Which statements are correct for the equation $(2y^2 +3x)dx + 2xydy = 0$?
(A) The differential equation is exact.
(B) It is possible to find an integrating factor so that the equation becomes exact.
(C) $x=-y^2$ is one possible solution.
(D) $x^2y^2 + x^2 = 1$ is one possible solution.
(E) None of the above

Consider the equation
$\frac{d^2y}{dx^2} - \frac{dy}{dx} + 2y = 2x - 2$
with the boundary conditions $y(0) = 0$, $y(\pi) = \pi$. Which statements are correct?
(A) The particular solution is $y_p = -x$.
(B) The general solution is $y = e^x \sin x + x$.
(C) There is a unique solution to this boundary-value problem.
(D) This differential equation is homogeneous.
(E) None of the above.

Which are the possible solutions of the differential equation
$y^2 \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$
(A) $y + \ln|y-1| = x$
(B) $2y + \ln|y-1| = 4x$
(C) $2y + \ln|2y-1| = 4x + 4$
(D) $y^2 = 2x$
(E) $y^2 = -2x+1$

Assume $y = \sum_{n=0}^\infty c_n x^n$ and solve $\frac{d^2y}{dx^2} - xy = 0$. Which statements are always true?
(A) $c_0 = 0$
(B) $c_1 = 0$
(C) $c_2 = 0$
(D) $c_n = \frac{c_{n-3}}{(n)(n-1)}$ for $n = 0, 1, 2, \dots$
(E) $c_n = \frac{c_{n-3}}{(n)(n-1)}$ for $n = 3, 4, 5, \dots$

Consider the equation $\frac{dy}{dt} = 1 - \sin t - \int_0^t y(\tau) d\tau$ with the initial condition $y(0) = 1$. Which statements about the function $y(t)$ and its Laplace transform $Y(s)$ are correct?
(A) $Y(s) = \frac{s+1}{s^2+1} \frac{1}{(s^2+1)^2}$
(B) $Y(s) = \frac{1}{s^2+1} \left( \frac{1}{s^2+1} \right)^2$
(C) $y(t) = \sin t - \frac{1}{2} t \sin t$
(D) $y(\pi) = 0$
(E) $y(\pi) = -1$

Which are the possible solutions of the given system?
$\begin{cases} \frac{dx}{dt} = 3x + 2y + 2e^t \\ \frac{dy}{dt} = -2x - y + e^t \end{cases}$
(A) $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1/2 \\ -1 \end{bmatrix} e^t$
(B) $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1/2 \\ -2 \end{bmatrix} e^t$
(C) $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ -2 \end{bmatrix} e^t$
(D) $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 1/2 \end{bmatrix} e^t$
(E) $\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1/2 \\ 1 \end{bmatrix} e^t$

Consider the wave equation $x^2 \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}$ for $-\infty < x < \infty, t > 0$ with initial conditions $u(x, 0) = f(x), \frac{\partial u}{\partial t}(x, 0) = 0$.
Which are the possible solutions?
(A) $u(x,t)=f(x-at)$
(B) $u(x,t)=f(x+at)$
(C) $u(x,t)=f(x-at)+f(x+at)$
(D) $u(x,t)=\frac{1}{2} [f(x-at)+f(x+at)]$
(E) None of the above.

Evaluate the Fourier transform
$F(\omega) = \int_{-\infty}^{\infty} e^{-r^2} e^{-i\omega t} dt$
(A) $\sqrt{\pi} t e^{-(\omega - \omega_0)^2}$
(B) $2\sqrt{\pi} t e^{-(\omega - \omega_0)^2}$
(C) $2\pi t e^{-(\omega - \omega_0)^2 / 2}$
(D) $2\sqrt{\pi} t e^{-(\omega - \omega_0)^2 / 2}$
(E) $\sqrt{2\pi} t e^{-(\omega - \omega_0)^2 / 2}$

Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system:
$\begin{bmatrix} 1 & -4 & 7 & g \\ 0 & 3 & -5 & h \\ -2 & 5 & -9 & k \end{bmatrix}$
(A) $2k+g+h = 0$
(B) $k+2g+3h = 0$
(C) $k+2g+h = 0$
(D) $3k+g+2h = 0$

Let $\mathbf{a}_1 = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}$, $\mathbf{a}_2 = \begin{bmatrix} -3 \\ -7 \\ 7 \end{bmatrix}$, and $\mathbf{b} = \begin{bmatrix} h \\ 1 \\ 4 \end{bmatrix}$. For what value(s) of h is $\mathbf{b}$ in the plane spanned by $\mathbf{a}_1$ and $\mathbf{a}_2$?
(A) 0
(B) -10
(C) 16
(D) -17
(E) 20

Determine the rank of the matrix in the following.
$A = \begin{bmatrix} 2 & 5 & -3 & -4 & 8 \\ 4 & 7 & -4 & -3 & 9 \\ 6 & 9 & -5 & -2 & 4 \\ 0 & -9 & 6 & 5 & -6 \end{bmatrix}$
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Matrix A can be written in the form A=LU, where L is an m x m lower triangular matrix with 1's on the diagonal and U is an m x n echelon form of A. Such a factorization is called an LU factorization. Please find the L matrix for LU factorization of the following matrix A.
$A = \begin{bmatrix} 3 & -6 & 3 \\ 6 & -7 & 6 \\ -1 & 7 & 0 \end{bmatrix}$
(A) $\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1/3 & 1 & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1/3 & 2 & 1 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 2/5 & 2 & 1 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & 1 & 1 \\ 3 \end{bmatrix}$
(E) $\begin{bmatrix} 1 & 0 & 0 \\ 2 & -5 & 1 \\ 2 & 1 & 1 \end{bmatrix}$

Find the determinant for the following matrix
$\begin{bmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \end{bmatrix}$
(A) -5
(B) 5
(C) 1
(D) -1
(E) 0

Find the dimension of the subspace H =
$\left\{ \begin{bmatrix} a-3b+6c \\ 5a+4d \\ b-2c-d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}$
in $\mathbb{R}^3$.
(A) 2
(B) 3
(C) 4
(D) 1
(E) 0

Let $\mathbf{b}_1 = \begin{bmatrix} 1 \\ -3 \end{bmatrix}$, $\mathbf{b}_2 = \begin{bmatrix} -2 \\ 7 \end{bmatrix}$, $\mathbf{c}_1 = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$, $\mathbf{c}_2 = \begin{bmatrix} 3 \\ 9 \end{bmatrix}$, and consider the bases for $\mathbb{R}^2$ given by $B = \{\mathbf{b}_1, \mathbf{b}_2\}$ and $C = \{\mathbf{c}_1, \mathbf{c}_2\}$. Find the change-of-coordinates matrix from $C$ to $B$.
(A) $\begin{bmatrix} 5 & 3 \\ 6 & 4 \end{bmatrix}$
(B) $\begin{bmatrix} 5 & 3 \\ 2 & 6 \end{bmatrix}$
(C) $\begin{bmatrix} 2 & 7 \\ 1 & 4 \end{bmatrix}$
(D) $\begin{bmatrix} 5 & 7 \\ 3 & 4 \end{bmatrix}$