Problem 1
I. Obtain the general solution of the following differential equation: $$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } + y = x ^ { 3 }$$
II. Obtain the general solution of the following differential equation: $$x ^ { 2 } \frac { d y } { d x } - x ^ { 2 } y ^ { 2 } + x y + 1 = 0$$ Note that $y = \frac { 1 } { x }$ is a particular solution.
III. Let $I _ { n }$ be defined by: $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { n } x \, d x$$ where $n$ is a non-negative integer.

1. Calculate $I _ { 0 } , I _ { 1 }$, and $I _ { 2 }$.
2. Calculate $I _ { n }$ for $n \geq 2$.

Problem 2
I. Answer the following questions about the matrix $\boldsymbol { P }$: $$\boldsymbol { P } = \left( \begin{array} { c c c } 0 & 0 & \frac { 3 } { 2 } \\ 2 & 0 & 0 \\ 0 & \frac { 1 } { 3 } & 0 \end{array} \right)$$

1. Obtain all eigenvalues of the matrix $\boldsymbol { P }$ and the corresponding eigenvectors with unit norms.
2. Obtain $P ^ { 2 }$ and $P ^ { 3 }$.

II. Let $\boldsymbol { A }$ be the real matrix given by the block diagonal matrix: $$\boldsymbol { A } = \left( \begin{array} { c c c c c } 0 & 0 & c & 0 & 0 \\ a & 0 & 0 & 0 & 0 \\ 0 & b & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & e \\ 0 & 0 & 0 & d & 0 \end{array} \right)$$ Express succinctly the necessary and sufficient condition on $a , b , c , d$, and $e$, such that there exists a positive integer $m$ for which $\boldsymbol { A } ^ { m }$ is the identity matrix (proof is not required).
III. The matrix $M$ is a square matrix of order 12 with all elements taking either 0 or 1, such that each row and column has exactly one element being 1. Let $k _ { 0 }$ be the minimum value of the positive integer $k$ such that $M ^ { k }$ is the identity matrix. For all possible matrices $M$, give the maximum value of $k _ { 0 }$ (proof is not required).

Problem 3
In the following, $z$ denotes a complex number and $i$ is the imaginary unit. The real part and the imaginary part of $z$ are denoted by $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$, respectively.
I. Answer the following questions.

1. Give the solutions of $z ^ { 5 } = 1$ in polar form. Plot the solutions on the complex plane.
2. The mapping $f$ is defined by $f : z \mapsto f ( z ) = \exp ( i z )$. Plot the image of the region $D = \{ z : \operatorname { Re } ( z ) \geq 0, 1 \geq \operatorname { Im } ( z ) \geq 0 \}$ under $f$ on the complex plane.
3. Find the residue of the function $z ^ { 2 } \exp \left( \frac { 1 } { z } \right)$ at $z = 0$.

II. Consider the complex function: $f ( z ) = \frac { ( \log z ) ^ { 2 } } { ( z + a ) ^ { 2 } }$, where $a$ is a positive real number. The closed path $C$ shown in Figure 3.1 is defined by $C = C _ { + } + C _ { R } + C _ { - } + C _ { r }$, where $R > a > r > 0$. Here, $\log z$ takes the principal value on the path $C _ { + }$. Answer the following questions.

1. Apply the residue theorem to calculate the contour integral $\oint _ { C } f ( z ) \, d z$.
2. Use the result of Question II.1 to calculate the integral: $\int _ { 0 } ^ { \infty } \frac { \log x } { ( x + a ) ^ { 2 } } \, d x$.

Problem 4
Answer the following questions on shapes in the three-dimensional orthogonal coordinate system $xyz$.
I. Consider the surface $S _ { 1 }$ represented by the equation $x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } = 0$. Find the equations expressed in $x , y$, and $z$ of the normal line and the tangent plane $T$ to the surface $S _ { 1 }$ at the point $\mathrm { A } ( 2, 0, 2 )$.
II. Consider the surface $S _ { 2 }$ represented by the following set of equations with the parameters $u$ and $v$: $$\left\{ \begin{array} { l } x = \frac { 1 } { \sqrt { 2 } } \cosh u \cos v \\ y = \frac { 1 } { 2 } \cosh u \sin v - \frac { 1 } { \sqrt { 2 } } \sinh u \\ z = \frac { 1 } { 2 } \cosh u \sin v + \frac { 1 } { \sqrt { 2 } } \sinh u \end{array} \right.$$ where $u$ and $v$ are real numbers, and $0 \leq v < 2 \pi$.
Let $S _ { 3 }$ be the surface obtained by rotating the surface $S _ { 2 }$ around the $x$-axis by $- \pi / 4$. Here, the positive direction of rotation is the direction of the semi-circular arrow on the $yz$-plane shown in Figure 4.1.
Answer the following questions.

1. Find the matrix $\boldsymbol { R }$ that represents the linear transformation rotating a shape around the $x$-axis by $- \pi / 4$.
2. Find an equation expressed in $x , y$, and $z$ for the surface $S _ { 3 }$.
3. Find an equation expressed in $x , y$, and $z$ for the surface $S _ { 2 }$.

III. Consider the solid $V$ that is enclosed by the surface $S _ { 3 }$ obtained in Question II.2 and by the two planes $z = 1$ and $z = - 1$. Answer the following questions.

1. Calculate the area of the cross section obtained by cutting the solid $V$ with the $xz$-plane.
2. Calculate the area of the cross section obtained by cutting the solid $V$ with the plane $T$ obtained in Question I.

Problem 5
Consider the continuously differentiable function $f ( x )$ of the real variable $x$. Let $f ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$. $f ( x )$, its derivative $f ^ { \prime } ( x )$, and $x f ( x )$ are absolutely integrable. The Fourier transform of the function $f ( x )$ is denoted by $\mathcal { F } \{ f ( x ) \} ( u )$ or equivalently by $\hat { f } ( u )$, and defined by $$\mathcal { F } \{ f ( x ) \} ( u ) = \hat { f } ( u ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } f ( x ) \exp ( - i u x ) \, d x \tag{1}$$ where $u$ is a real variable and $i$ is the imaginary unit. The Fourier transform is defined in the same way for other functions.
I. Express $\mathcal { F } \left\{ f ^ { \prime } ( x ) \right\} ( u )$ in terms of $\hat { f } ( u )$ and $u$.
II. Express $\frac { d \hat { f } ( u ) } { d u }$ in terms of $\mathcal { F } \{ x f ( x ) \} ( u )$.
III. Let the function $f ( x ) = \exp \left( - a x ^ { 2 } \right)$, where $a$ is a positive real constant $( a > 0 )$. The following relation holds for $f ( x )$: $$f ^ { \prime } ( x ) = - 2 a x f ( x ) \tag{2}$$ Apply the Fourier transform on both sides of Eq. (2) to obtain a first-order ordinary differential equation in $\hat { f } ( u )$. Solve this ordinary differential equation to obtain $\hat { f } ( u )$. Note that the integration constant in the solution of this ordinary differential equation can be obtained by calculating $\hat { f } ( 0 )$ with the help of Eq. (1) and the value of the following improper integral: $$\int _ { - \infty } ^ { \infty } \exp \left( - a x ^ { 2 } \right) d x = \sqrt { \frac { \pi } { a } } \tag{3}$$
IV. Consider the function $h ( x , t )$ of the real variables $x$ and $t$. Let $h ( x , t )$ be defined for $- \infty < x < \infty$ and $t \geq 0$, and satisfy the following partial differential equation: $$\frac { \partial h ( x , t ) } { \partial t } = \frac { \partial ^ { 2 } h ( x , t ) } { \partial x ^ { 2 } } \quad ( t > 0 ) \tag{4}$$ given the initial condition $$h ( x , 0 ) = \exp \left( - a x ^ { 2 } \right) \quad ( a > 0 ) \tag{5}$$

1. Apply the Fourier transform with respect to the variable $x$ on both sides of the partial differential equation (4) to obtain an ordinary differential equation with $\hat { h } ( u , t ) \equiv \frac { 1 } { \sqrt { 2 \pi } } \int _ { - \infty } ^ { \infty } h ( x , t ) \exp ( - i u x ) \, d x$ and the independent variable $t$.
2. By solving the ordinary differential equation found in Question IV.1, obtain $\hat { h } ( u , t )$.
3. Use the inverse Fourier transform with respect to the variable $u$ to obtain a solution $h ( x , t )$ satisfying Eq. (4) and Eq. (5).

V. Consider the continuous function $g ( x )$ and its Fourier transform $\hat { g } ( u )$. Let $g ( x ) \rightarrow 0$ as $| x | \rightarrow \infty$ and $g ( x )$ be absolutely integrable. The convolution of the functions $f ( x )$ and $g ( x )$ is defined by $$( f * g ) ( x ) \equiv \int _ { - \infty } ^ { \infty } f ( y ) g ( x - y ) \, d y \tag{6}$$

1. Express $\mathcal { F } \{ ( f * g ) ( x ) \} ( u )$ in terms of $\hat { f } ( u )$ and $\hat { g } ( u )$.
2. Here, the function $h ( x , t )$ satisfies Eq. (4), given the initial condition $h ( x , 0 ) = g ( x )$. Use the result of Question V.1 to find an integral representation of a solution $h ( x , t )$, where $t > 0$.

Problem 6
Consider $n$ random variables $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$ that can take the values 0 and 1. Here, $n$ is an integer greater than or equal to 4. The probability of an event $A$ is denoted by $P ( A )$, and the conditional probability of the event $A$ given an event $B$ is denoted by $P ( A \mid B )$. The intersection between the event $A$ and the event $B$ is denoted by $A \wedge B$. Answer the following questions.
I. Let us assume that the $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$ are independent. In addition, assume that each $X _ { k } \quad ( k = 1, 2 , \cdots , n )$ takes the value 1 with the probability $p$ and the value 0 with the probability $1 - p$, i.e., $P \left( X _ { k } = 1 \right) = p$ and $P \left( X _ { k } = 0 \right) = 1 - p$.

1. Find the expected value and the variance of the sum of the $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$.
2. The random variables $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$ are arranged in the row $X _ { n } \cdots X _ { 2 } X _ { 1 }$. Let $Y$ be the integer value obtained by regarding that row as an $n$-digit binary number. For example, in the case that $n = 4$, $Y = 5$ when the row $X _ { 4 } X _ { 3 } X _ { 2 } X _ { 1 }$ is 0101, and $Y = 13$ when the row $X _ { 4 } X _ { 3 } X _ { 2 } X _ { 1 }$ is 1101. $Y$ is a random variable that takes integer values from 0 to $2 ^ { n } - 1$. Obtain the expected value and variance of $Y$.

II. The values of the random variables $X _ { 1 } , X _ { 2 } , \cdots , X _ { n }$ are obtained sequentially according to the following steps. First, $X _ { 1 }$ takes the value 1 with the probability $p$ and the value 0 with the probability $1 - p$. Then, $X _ { k } \quad ( k = 2, 3 , \cdots , n )$ takes the same value as $X _ { k - 1 }$ with the probability $q$ and the value different from $X _ { k - 1 }$ with the probability $1 - q$, i.e., $P \left( X _ { k } = 1 \mid X _ { k - 1 } = 1 \right) = P \left( X _ { k } = 0 \mid X _ { k - 1 } = 0 \right) = q$ and $P \left( X _ { k } = 1 \mid X _ { k - 1 } = 0 \right) = P \left( X _ { k } = 0 \mid X _ { k - 1 } = 1 \right) = 1 - q$.

1. Let $P \left( X _ { k } = 1 \right)$ be represented by $r _ { k }$, where $k$ is an integer varying from 1 to $n$. Derive a recurrence equation for $r _ { k }$. Solve this recurrence equation to express $r _ { k }$ with $p , q$, and $k$.
2. Obtain the probability $P \left( X _ { 1 } = 1 \wedge X _ { 2 } = 0 \wedge X _ { 3 } = 1 \wedge X _ { 4 } = 0 \right)$.
3. Obtain the probability $P \left( X _ { 3 } = 1 \mid X _ { 1 } = 0 \wedge X _ { 2 } = 1 \wedge X _ { 4 } = 1 \right)$.