2023

Problem 1

Answer the following questions.

(1) The function $f(x,y)$ with real variables $x,y$ is defined as follows:

$$f(x,y)=\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& x& y\end{array}\right|.$$

Show that the set of solutions of the equation $f(x,y)=0$ is a line passing through two points $(x_1,y_1),(x_2,y_2)$ on the $xy$ plane, where $x_1\ne x_2$.

(2) Find the value of the determinant

$$\left|\begin{array}{ccc}1& x_1& x_1^2\\ 1& x_2& x_2^2\\ 1& x_3& x_3^2\end{array}\right|$$

in factored form.

(3) Show that there is a unique curve $y=a_0+a_{1}x+a_2{x}^2$ passing through three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ on the $xy$ plane, where $a_0,a_1,a_2$ are constants and $x_1,x_2,x_3$ are all distinct.

(4) The curve in (3) can be represented in the form $y=c_1{y}_1+c_2{y}_2+c_3{y}_3$, where each of $c_1,c_2,c_3$ does not depend on $y_1,y_2,y_3$. Find $c_1,c_2,c_3$.

(5) Let us represent a curve $y=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4$ passing through five points $(x_1,y_1),\dots ,(x_5,y_5)$ on the $xy$ plane in the form $y=c_1{y}_1+\cdots +c_5{y}_5$, where each of $c_1,\dots ,c_5$ does not depend on $y_1,\dots ,y_5$ and $x_1,\dots ,x_5$ are all distinct. Find $c_1$.

---

回答以下问题。

(1) 带有实变量 $x,y$ 的函数 $f(x,y)$ 定义如下：

$$f(x,y)=\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& x& y\end{array}\right|.$$

证明方程 $f(x,y)=0$ 的解集是通过 $xy$ 平面上两点 $(x_1,y_1),(x_2,y_2)$ 的一条直线，其中 $x_1\ne x_2$。

(2) 求下列行列式的值

$$\left|\begin{array}{ccc}1& x_1& x_1^2\\ 1& x_2& x_2^2\\ 1& x_3& x_3^2\end{array}\right|$$

的因式分解形式。

(3) 证明存在唯一的曲线 $y=a_0+a_{1}x+a_2{x}^2$，通过 $xy$ 平面上的三点 $(x_1,y_1),(x_2,y_2),(x_3,y_3)$，其中 $a_0,a_1,a_2$ 是常数，且 $x_1,x_2,x_3$ 彼此不同。

(4) (3) 中的曲线可以表示为 $y=c_1{y}_1+c_2{y}_2+c_3{y}_3$ 的形式，其中 $c_1,c_2,c_3$ 均不依赖于 $y_1,y_2,y_3$。求 $c_1,c_2,c_3$。

(5) 设曲线 $y=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4$ 通过 $xy$ 平面上的五个点 $(x_1,y_1),\dots ,(x_5,y_5)$，并表示为 $y=c_1{y}_1+\cdots +c_5{y}_5$ 的形式，其中 $c_1,\dots ,c_5$ 均不依赖于 $y_1,\dots ,y_5$，且 $x_1,\dots ,x_5$ 彼此不同。求 $c_1$。

---

Problem 2

Let $t$ be a real independent variable, and let $x(t)$ and $y(t)$ be real-valued functions. Answer the following questions.

(1) Find all solutions of the following ordinary differential equation

$$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}x}{\mathrm{d}t}+x=\cos (t),$$

which are bounded when $t\to -\infty$.

(2) Find all solutions $x(t)$ and $y(t)$ of the following ordinary differential equations

$$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}x}{\mathrm{d}t}+x-y=\cos (t),$$ $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}y}{\mathrm{d}t}+y-x=0,$$

which are bounded when $t\to -\infty$.

(3) By converting the following ordinary differential equation

$$e^{-t}{x}^2-2\frac{\mathrm{d}x}{\mathrm{d}t}+x=0$$

to a linear ordinary differential equation with an appropriate change of variable, find the solution $x(t)$ that satisfies $x(0)=\frac{1}{2}$.

---

设 $t$ 为实独立变量，$x(t)$ 和 $y(t)$ 为实值函数。回答以下问题。

(1) 求解下列常微分方程的所有解

$$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}x}{\mathrm{d}t}+x=\cos (t),$$

这些解在 $t\to -\infty$ 时是有界的。

(2) 求解下列常微分方程的所有解 $x(t)$ 和 $y(t)$

$$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}x}{\mathrm{d}t}+x-y=\cos (t),$$ $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{t}^2}+2\frac{\mathrm{d}y}{\mathrm{d}t}+y-x=0,$$

这些解在 $t\to -\infty$ 时是有界的。

(3) 通过对下列常微分方程

$$e^{-t}{x}^2-2\frac{\mathrm{d}x}{\mathrm{d}t}+x=0$$

进行适当的变量变换以将其转换为线性常微分方程，求解满足 $x(0)=\frac{1}{2}$ 的 $x(t)$。

---

Problem 3

Let us randomly place circle stones $◯$ and square stones $\square$ one by one in a line from left to right. The circle and square stones are placed with probability $1-q$ and $q$, respectively, according to the independent and identical distribution, where $0<q<1$. The placement stops right after $M$ square stones are placed in a row, where $M$ is a positive integer. We show examples of the lines for $M=4$ as follows.

$$\text{line 1: }◯\square \square \square \square$$ $$\text{line 2: }\square ◯\square ◯◯\square \square \square \square$$

Let $L$ be a random variable which represents the number of the stones after stopping the placement. For the case of the lines shown above, $L=5$ and $L=9$ for lines 1 and 2, respectively.

Here, we consider intermediate states during the placement. Let $k$ be a non-negative integer and let $C_k$ be a state of a line where there are $k$ square stones in a row from the right end. For instance, we consider the following lines for $M=4$.

$$\text{line 3: }◯\square \square \square ◯◯\square \square$$ $$\text{line 4: }\square ◯\square ◯◯$$

Since we are considering the case of $M=4$, lines 3 and 4 are not stopped yet. Line 3 is in state $C_2$ since there are 2 square stones in a row from the right end. Line 4 is in state $C_0$ since there is no square stone at the right end. Let $a_{kn}$ be the probability that the stopping condition is met after placing $n$ stones starting from state $C_k$, where $n$ is a non-negative integer. We define the following generating function $A_k(t)$ for $a_{kn}$.

$$A_k(t)=\sum _{n=0}^{\infty }{t}^n{a}_{kn}$$

Answer the following questions.

(1) Calculate the mean and variance of $L$ for $M=1$.

(2) Obtain the recurrence relation that $A_k(t)$ satisfies.

(3) Obtain $A_k(t)$ as a function of $q$, $M$, $t$, and $k$.

(4) Calculate the mean of $L$.

---

我们从左到右逐一随机放置圆形石头 $◯$ 和方形石头 $\square$。圆形和方形石头分别以 $1-q$ 和 $q$ 的概率放置，遵循独立且相同的分布，其中 $0<q<1$。在放置了 $M$ 个方形石头连续排列之后，放置将停止，其中 $M$ 是一个正整数。以下是 $M=4$ 时的示例行。

$$\text{line 1: }◯\square \square \square \square$$ $$\text{line 2: }\square ◯\square ◯◯\square \square \square \square$$

设 $L$ 为一个随机变量，表示停止放置后石头的数量。在上述行的情况下，线 1 的 $L=5$，线 2 的 $L=9$。

在这里，我们考虑放置过程中的中间状态。设 $k$ 为一个非负整数，$C_k$ 为一个状态，表示从右端起有 $k$ 个方形石头连续排列。例如，我们考虑 $M=4$ 时的以下行。

$$\text{line 3: }◯\square \square \square ◯◯\square \square$$ $$\text{line 4: }\square ◯\square ◯◯$$

由于我们考虑的是 $M=4$ 的情况，线 3 和线 4 尚未停止。线 3 处于状态 $C_2$，因为从右端起有 2 个方形石头连续排列。线 4 处于状态 $C_0$，因为右端没有方形石头。设 $a_{kn}$ 为在从状态 $C_k$ 开始放置 $n$ 个石头后满足停止条件的概率，其中 $n$ 为非负整数。我们定义以下 $a_{kn}$ 的生成函数 $A_k(t)$。

$$A_k(t)=\sum _{n=0}^{\infty }{t}^n{a}_{kn}$$

回答以下问题。

(1) 计算 $M=1$ 时 $L$ 的均值和方差。

(2) 求出 $A_k(t)$ 满足的递推关系。

(3) 将 $A_k(t)$ 表示为 $q$, $M$, $t$ 和 $k$ 的函数。

(4) 计算 $L$ 的均值。