Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
Show that if $B$ has an eigenvalue of the form $\lambda = i\frac{2k\pi}{mT}$ with $k \in \mathbb{Z}$ and $m \in \mathbb{N}^*$, then (2) has a non-zero $mT$-periodic solution.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
Suppose that there exists $m \in \mathbb{N}^*$ such that (2) has a non-zero $mT$-periodic solution. Show that $\exp(TB)$ has an eigenvalue that is an $m$-th root of unity.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix such that $M(t+T) = M(t)\exp(TB)$ for all $t$.
Show that if $B$ has an eigenvalue of the form $\lambda = i\frac{2k\pi}{mT}$ with $k \in \mathbb{Z}$ and $m \in \mathbb{N}^*$, then (2) has a non-zero $mT$-periodic solution.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix such that $M(t+T) = M(t)\exp(TB)$ for all $t$.
Suppose that there exists $m \in \mathbb{N}^*$ such that (2) has a non-zero $mT$-periodic solution. Show that $\exp(TB)$ has an eigenvalue that is an $m$-th root of unity.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ In this question, we suppose that (2) has a $T'$-periodic solution $X$ with $T' \notin \mathbb{Q} T$.
Show that for all $t \in \mathbb{R}$ and $u \in \mathbb{R}$, we have $$A(u) X(t) = A(t) X(t).$$ One may use without proof the fact that if $G$ is a subgroup of $(\mathbb{R}, +)$ which is not of the form $\mathbb{Z}a$ for $a \in \mathbb{R}$, then $G$ is dense in $\mathbb{R}$.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. Let the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic.
We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix such that $M(t+T) = M(t)\exp(TB)$ for all $t$. Consider the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic. We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.

Solve the differential system $$\left\{ \begin{array}{l} x'(t) = x(t) - \cos(t) y(t) \\ y'(t) = \cos(t) x(t) + y(t) \end{array} \right.$$ and determine its normal form (see question 16d).

Solve the differential system $$\left\{\begin{array}{l} x'(t) = x(t) - \cos(t) y(t) \\ y'(t) = \cos(t) x(t) + y(t) \end{array}\right.$$ and determine its normal form (see question 16d).

Justify that there exists a unique solution $u$ to the Cauchy problem $\left( C _ { \ell } \right)$, give its expression and draw its variation table.
$$\left( C _ { \ell } \right) : \left\{ \begin{array} { l } u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) ) \\ u ( 0 ) = 0 \end{array} . \right.$$

Show that there exists a unique constant solution of equation $\left( E _ { \ell } \right)$, denoted $\gamma \in \mathbf { R }$, and verify that the solution $u$ found in question 1 satisfies
$$\lim _ { x \rightarrow + \infty } u ( x ) = \gamma .$$
where $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $g$. a. Show that $g ( x )$ is well defined for all $x \geqslant 0$. b. Show that $g$ is of class $C ^ { 2 }$ on $] 0 , \infty [$.
For this you may use the change of variable $u = t + x$ and express $g$ in terms of the functions $C : x \mapsto \int _ { x } ^ { \infty } \frac { \cos u } { u } \mathrm {~d} u$ and $S : x \mapsto \int _ { x } ^ { \infty } \frac { \sin u } { u } \mathrm {~d} u$. c. Determine a linear second-order differential equation satisfied by $f$.

The solution of the differential equation $\frac { d y } { d x } + \frac { y } { 2 } \sec x = \frac { \tan x } { 2 y }$, where $0 \leq x < \frac { \pi } { 2 }$ and $y ( 0 ) = 1$, is given by
(1) $y ^ { 2 } = 1 + \frac { x } { \sec x + \tan x }$
(2) $y = 1 + \frac { x } { \sec x + \tan x }$
(3) $y = 1 - \frac { x } { \sec x + \tan x }$
(4) $y ^ { 2 } = 1 - \frac { x } { \sec x + \tan x }$

Problem 1
I. Find the general solution of the following differential equation:
$$\frac { d ^ { 4 } y } { d x ^ { 4 } } - 2 \cdot \frac { d ^ { 3 } y } { d x ^ { 3 } } + 2 \frac { d y } { d x } - y = 9 e ^ { - 2 x }$$
Here, $e$ denotes the base of the natural logarithm.
II. Find the value of the following integral:
$$\int _ { 0 } ^ { 1 } x ^ { m } ( \log x ) ^ { n } d x$$
Here, $m$ and $n$ are non-negative integers.
III. We define $I ( m )$ as
$$I ( m ) \equiv \int _ { 0 } ^ { 1 } x ^ { m } \arccos x \, d x$$
Here, $m$ is a non-negative integer. Use the principal values of inverse trigonometric functions.

1. Find the value of $I ( 0 )$.
2. Find the value of $I ( 1 )$.
3. Express $I ( m )$ in terms of $m$ and $I ( m - 2 )$ when $m \geq 2$.
4. Find the value of $I ( m )$.

Problem 5
I. A function $f ( x )$ is continuous and defined on the interval $0 \leq x \leq \pi$. If $f ( x )$ is extended to the interval $- \pi \leq x \leq \pi$ as an odd function, it can be expanded in the following Fourier sine series:
$$\begin{aligned} & f ( x ) \doteq \sum _ { n = 1 } ^ { \infty } \left( b _ { n } \sin n x \right) \\ & b _ { n } = \frac { 2 } { \pi } \int _ { 0 } ^ { \pi } f ( x ) \sin n x \, d x \quad ( n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 ) = f ( \pi ) = 0$.

1. Find the Fourier sine series for the following function $f ( x )$: $$f ( x ) = x ( \pi - x ) \quad ( 0 \leq x \leq \pi )$$
2. Derive the following equation using the result obtained in Question I.1, $$\frac { 1 } { 1 ^ { 3 } } - \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 5 ^ { 3 } } - \frac { 1 } { 7 ^ { 3 } } + \cdots = \frac { \pi ^ { 3 } } { 32 }$$

II. A two-variable function $f ( x , y )$ is continuous and defined in the region $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. Using a similar method to Question I, $f ( x , y )$ can be expanded in the following double Fourier sine series:
$$\begin{aligned} & f ( x , y ) = \sum _ { m = 1 } ^ { \infty } \sum _ { n = 1 } ^ { \infty } \left( B _ { m n } \sin m x \sin n y \right) \\ & B _ { m n } = \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } f ( x , y ) \sin m x \sin n y \, d x \, d y \quad ( m , n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 , y ) = f ( \pi , y ) = f ( x , 0 ) = f ( x , \pi ) = 0$.

1. Find the double Fourier sine series for the following function $f ( x , y )$: $$f ( x , y ) = x ( \pi - x ) \sin y \quad ( 0 \leq x \leq \pi , 0 \leq y \leq \pi )$$
2. Function $u ( x , y , t )$ is defined in the region $0 \leq x \leq \pi , 0 \leq y \leq \pi$ and $t \geq 0$. Obtain the solution for the following partial differential equation of $u ( x , y , t )$ by the method of separation of variables: $$\frac { \partial u } { \partial t } = c ^ { 2 } \left( \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } \right)$$ where $c$ is a positive constant and the following boundary and initial conditions apply: $$\begin{aligned} & u ( 0 , y , t ) = u ( \pi , y , t ) = u ( x , 0 , t ) = u ( x , \pi , t ) = 0 \\ & u ( x , y , 0 ) = x ( \pi - x ) \sin y \end{aligned}$$

Problem 1

I. Find the value of the following definite integral:
$$I = \int _ { 2 } ^ { 4 } \frac { d x } { \sqrt { ( x - 2 ) ( 4 - x ) } }$$
II. Find the general solution and the singular solution of the following differential equation:
$$y = x \frac { d y } { d x } + \frac { d y } { d x } + \left( \frac { d y } { d x } \right) ^ { 2 }$$
III. Find the general solution of the following differential equation:
$$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } - 8 y = x ^ { 2 }$$

A real-valued function $u ( x , t )$ is defined in $0 \leq x \leq 1$ and $t \geq 0$. Here, $x$ and $t$ are independent. Suppose solving the following partial differential equation:
$$\frac { \partial u } { \partial t } = \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } ,$$
under the following conditions:
$$\begin{aligned} \text { Boundary condition : } & u ( 0 , t ) = u ( 1 , t ) = 0 , \\ \text { Initial condition : } & u ( x , 0 ) = x - x ^ { 2 } . \end{aligned}$$
Since the constant function $u ( x , t ) = 0$ is obviously a solution of the partial differential equation, consider the other solutions. Answer the following questions.
(1) Calculate the following expression, where $n$ and $m$ are positive integers.
$$\int _ { 0 } ^ { 1 } \sin ( n \pi x ) \sin ( m \pi x ) \mathrm { d } x$$
(2) Suppose $u ( x , t ) = \xi ( x ) \tau ( t )$, where $\xi ( x )$ is a function only of $x$ and $\tau ( t )$ is a function only of $t$. Express the ordinary differential equations for $\xi$ and $\tau$ using an arbitrary constant $C$. You may use that $f ( x )$ and $g ( t )$ are constant functions when $f ( x )$ and $g ( t )$ satisfy $f ( x ) = g ( t )$ for arbitrary $x$ and $t$.
(3) Solve the ordinary differential equations in question (2). Next, show that a solution of partial differential equation $(*)$ which satisfies the boundary condition is given by the following $u _ { n } ( x , t )$, and express $\alpha$ and $\beta$ using a positive integer $n$.
$$u _ { n } ( x , t ) = e ^ { \alpha t } \sin ( \beta x )$$
(4) The solution of partial differential equation $(*)$ which satisfies the boundary and initial conditions is represented by the linear combination of $u _ { n } ( x , t )$ as shown below. Obtain $c _ { n }$. You may use the result of question (1).
$$u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } u _ { n } ( x , t )$$

Problem 5

The Laplace transform $F ( s ) = L [ f ( t ) ]$ of a function $f ( t )$, where $t \geq 0$, is defined as
$$F ( s ) = \int _ { 0 } ^ { \infty } f ( t ) e ^ { - s t } d t$$
Here, $s$ is a complex number, and $e$ is the base of the natural logarithm. Answer the following questions. Show the derivation process with your answer. I. Prove the following relations:

1. $L \left[ t ^ { n } \right] = \frac { n ! } { s ^ { n + 1 } }$, where $n$ is a natural number.
2. $L \left[ \frac { d f ( t ) } { d t } \right] = s F ( s ) - f ( 0 )$, where $f ( t )$ is a differentiable function.
3. $L \left[ e ^ { a t } f ( t ) \right] = F ( s - a )$, where $a$ is a real number.

II. Solve the following differential equation using a Laplace transformation for $t \geq 0$:
$$t \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } } + ( 1 + 3 t ) \frac { d f ( t ) } { d t } + 3 f ( t ) = 0 , \quad f ( 0 ) = 1 , \left. \quad \frac { d f } { d t } \right| _ { t = 0 } = - 3$$
You can use the relation $L [ t f ( t ) ] = - \frac { d } { d s } F ( s )$, if necessary. III. The point $\mathrm { P } ( x ( t ) , y ( t ) )$, which satisfies the following simultaneous differential equations, passes through the point $( a , b )$ when $t = 0$. $a$ and $b$ are real numbers.
$$\left\{ \begin{array} { l } \frac { d x ( t ) } { d t } = - x ( t ) \\ \frac { d y ( t ) } { d t } = x ( t ) - 2 y ( t ) \end{array} \right.$$

1. Solve the simultaneous differential equations using a Laplace transformation for $t \geq 0$.
2. Express the relation between $x$ and $y$ by eliminating $t$ from the solution of III. 1.
3. For both $( a , b ) = ( 1,1 )$ and $( - 1,1 )$, draw the trajectories of point P when $t$ varies continuously from 0 to infinity.

I. Find the general solutions of the following differential equations.

1. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = e ^ { x } \cos x$
2. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { 1 } { x } \frac { d y } { d x } + \frac { 4 } { x ^ { 2 } } y = \left( \frac { 2 \log x } { x } \right) ^ { 2 }$

II. Answer the following questions for the partial differential equation represented in Equation (3) and the boundary conditions represented in Equations (4)-(7):
$$\begin{aligned} & \frac { \partial ^ { 2 } u ( x , y ) } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u ( x , y ) } { \partial y ^ { 2 } } = 0 \quad ( 0 \leq x , 0 \leq y \leq 1 ) \\ & \left\{ \begin{array} { l } \lim _ { x \rightarrow + \infty } u ( x , y ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial y } \right| _ { y = 0 } = 0 \\ u ( x , 1 ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial x } \right| _ { x = 0 } = 1 + \cos \pi y \end{array} \right. \end{aligned}$$

1. Find the solution which satisfies Equations (3) and (4) in the form of $u ( x , y ) = X ( x ) \cdot Y ( y )$.
2. Find the solution satisfying Equations (5) and (6) for the solution of Question II.1.
3. Find the solution of the partial differential equation (3) satisfying all the boundary conditions given in Equations (4)-(7), using the solution of Question II.2.

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 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 $t > 0$.

I. Answer the following questions about the differential equation:
$$\cos x \frac { d ^ { 2 } y } { d x ^ { 2 } } - \sin x \frac { d y } { d x } - \frac { y } { \cos x } = 0 \quad \left( - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \right) .$$

1. A particular solution of Eq. (1) is of the form of $y = ( \cos x ) ^ { m }$ (m is a constant). Find the constant $m$.
2. Find the general solution of Eq. (1), using the solution of Question I.1.

II. Find the value of the following integral:
$$I = \int _ { 1 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 4 } + 2 x ^ { 2 } - 1 } d x$$
Note that, for a positive constant $\alpha$, the relation $\int _ { 0 } ^ { \infty } e ^ { - \alpha x ^ { 2 } } d x = \frac { 1 } { 2 } \sqrt { \frac { \pi } { \alpha } }$ holds.
III. Express the general solution of the following differential equation in the form of $f ( x , y ) = C$ ( $C$ is a constant) using an appropriate function $f ( x , y )$ :
$$\left( x ^ { 3 } y ^ { n } + x \right) \frac { d y } { d x } + 2 y = 0 \quad ( x > 0 , y > 0 )$$
where $n$ is an arbitrary real constant.

Consider a real-valued function $f(t)$ for a real variable $t$ defined for $0 \leq t < \infty$. The Laplace transform is defined as
$$\mathcal{L}[f(t)] = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, \mathrm{d}t \tag{1}$$
where $s$ is a complex variable whose real part is positive. Under the condition that the improper integral in the right-hand side does not diverge, answer the following questions.

1. When the conditions $$\lim_{t \rightarrow \infty} e^{-st} f(t) = 0 \quad \text{and} \quad \lim_{t \rightarrow \infty} e^{-st} f^{\prime}(t) = 0$$ are satisfied, show the following equation holds: $$\mathcal{L}\left[f^{\prime\prime}(t)\right] = -f^{\prime}(0) - s f(0) + s^{2} \mathcal{L}[f(t)]$$ Note that $f^{\prime}(t)$ and $f^{\prime\prime}(t)$ are defined as $$f^{\prime}(t) = \frac{\mathrm{d}f(t)}{\mathrm{d}t} \quad \text{and} \quad f^{\prime\prime}(t) = \frac{\mathrm{d}^{2}f(t)}{\mathrm{d}t^{2}}$$
   
2. Calculate the Laplace transform of $g(t) = e^{-at}\cos(\omega t)$ and $h(t) = e^{-at}\sin(\omega t)$ defined for $0 \leq t < \infty$ by showing derivation processes using Equation (1). Note that $a$ and $\omega$ are positive real numbers.
   
3. Solve the differential equation $$f^{\prime\prime}(t) + 6f^{\prime}(t) + 13f(t) = 0$$ where the initial values are $f(0) = 5$ and $f^{\prime}(0) = -11$.

Answer all the following questions.
I. Find the following limit value:
$$\lim _ { x \rightarrow 0 } \frac { b ^ { x } - c ^ { x } } { a x } \quad ( a , b , c > 0 )$$
II. Find the general solutions of the following differential equations.
$$\begin{aligned} & \text { 1. } \frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = \log x \quad ( x > 0 ) \\ & \text { 2. } \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 2 x ^ { 2 } + 2 x \end{aligned}$$
III. Let $a _ { n }$ be defined by
$$a _ { n } = \frac { n ! } { n ^ { n + \frac { 1 } { 2 } } e ^ { - n } }$$
where $n$ is a positive integer and $e$ is the base of natural logarithm. Find $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { a _ { n + 1 } }$. Note that the function $y = x ^ { - 1 } ( x > 0 )$ is convex downward.