Let $y = y(x)$ be the solution of the differential equation $\cos x\left(\log_e(\cos x)\right)^2 \mathrm{d}y + \left(\sin x - 3y\sin x\log_e(\cos x)\right)\mathrm{d}x = 0$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}$, then $y\left(\frac{\pi}{6}\right)$ is equal to:
(1) $\frac{1}{\log_e(3) - \log_e(4)}$
(2) $\frac{2}{\log_e(3) - \log_e(4)}$
(3) $\frac{1}{\log_e(4) - \log_e(3)}$
(4) $-\frac{1}{\log_e(4)}$

Let $y = y(x)$ be the solution of the differential equation $2\cos x \frac{\mathrm{d}y}{\mathrm{d}x} = \sin 2x - 4y \sin x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right) = 0$, then $y'\left(\frac{\pi}{4}\right) + y\left(\frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.

If $y = y ( x )$ is the solution of the differential equation, $\sqrt { 4 - x ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( \left( \sin ^ { - 1 } \left( \frac { x } { 2 } \right) \right) ^ { 2 } - y \right) \sin ^ { - 1 } \left( \frac { x } { 2 } \right) , - 2 \leq x \leq 2 , y ( 2 ) = \frac { \pi ^ { 2 } - 8 } { 4 }$, then $y ^ { 2 } ( 0 )$ is equal to

Let $y = f ( x )$ be the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { x y } { x ^ { 2 } - 1 } = \frac { x ^ { 6 } + 4 x } { \sqrt { 1 - x ^ { 2 } } } , - 1 < x < 1$ such that $f ( 0 ) = 0$. If $6 \int _ { - 1 / 2 } ^ { 1 / 2 } f ( x ) \mathrm { d } x = 2 \pi - \alpha$ then $\alpha ^ { 2 }$ is equal to $\_\_\_\_$

Q76. Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + y ^ { 2 } \right) e ^ { \tan x } d x + \cos ^ { 2 } x \left( 1 + e ^ { 2 \tan x } \right) d y = 0 , y ( 0 ) = 1$. Then $y \left( \frac { \pi } { 4 } \right)$ is equal to
(1) $\frac { 2 } { e }$
(2) $\frac { 2 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { 1 } { e ^ { 2 } }$

Q76. Let $y = y ( x )$ be the solution curve of the differential equation $\sec y \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 x \sin y = x ^ { 3 } \cos y , y ( 1 ) = 0$. Then $y ( \sqrt { 3 } )$ is equal to :
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 12 }$
(4) $\frac { \pi } { 4 }$

Q77. Let $y = y ( x )$ be the solution of the differential equation $\left( x ^ { 2 } + 4 \right) ^ { 2 } d y + \left( 2 x ^ { 3 } y + 8 x y - 2 \right) d x = 0$. If $y ( 0 ) = 0$, then $y ( 2 )$ is equal to
(1) $\frac { \pi } { 32 }$
(2) $2 \pi$
(3) $\frac { \pi } { 8 }$
(4) $\frac { \pi } { 16 }$

Q77. If $y = y ( x )$ is the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \sin ( 2 x ) , y ( 0 ) = \frac { 3 } { 4 }$, then $y \left( \frac { \pi } { 8 } \right)$ is equal to:
(1) $\mathrm { e } ^ { \pi / 8 }$
(2) $e ^ { \pi / 4 }$
(3) $e ^ { - \pi / 4 }$
(4) $e ^ { - \pi / 8 }$

Q77. Let $y = y ( x )$ be the solution of the differential equation $\left( 1 + x ^ { 2 } \right) \frac { d y } { d x } + y = e ^ { \tan ^ { - 1 } x } , y ( 1 ) = 0$. Then $y ( 0 )$ is
(1) $\frac { 1 } { 2 } \left( e ^ { \pi / 2 } - 1 \right)$
(2) $\frac { 1 } { 2 } \left( 1 - e ^ { \pi / 2 } \right)$
(3) $\frac { 1 } { 4 } \left( 1 - e ^ { \pi / 2 } \right)$
(4) $\frac { 1 } { 4 } \left( e ^ { \pi / 2 } - 1 \right)$

Q78. Let $y = y ( x )$ be the solution of the differential equation $\left( 2 x \log _ { e } x \right) \frac { d y } { d x } + 2 y = \frac { 3 } { x } \log _ { e } x , x > 0$ and $y \left( e ^ { - 1 } \right) = 0$. Then, $y ( e )$ is equal to
(1) $- \frac { 3 } { \mathrm { e } }$
(2) $- \frac { 3 } { 2 e }$
(3) $- \frac { 2 } { 3 e }$
(4) $- \frac { 2 } { \mathrm { e } }$

Q88. Let the solution $y = y ( x )$ of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } - y = 1 + 4 \sin x$ satisfy $y ( \pi ) = 1$. Then $y \left( \frac { \pi } { 2 } \right) + 10$ is equal to $\_\_\_\_$

Q88. If the solution $y ( x )$ of the given differential equation $\left( \mathrm { e } ^ { y } + 1 \right) \cos x \mathrm {~d} x + \mathrm { e } ^ { y } \sin x \mathrm {~d} y = 0$ passes through the point $\left( \frac { \pi } { 2 } , 0 \right)$, then the value of $\mathrm { e } ^ { y \left( \frac { \pi } { 6 } \right) }$ is equal to $\_\_\_\_$

Q88. For a differentiable function $f : \mathbb { R } \rightarrow \mathbb { R }$, suppose $f ^ { \prime } ( x ) = 3 f ( x ) + \alpha$, where $\alpha \in \mathbb { R } , f ( 0 ) = 1$ and $\lim _ { x \rightarrow - \infty } f ( x ) = 7$. Then $9 f \left( - \log _ { \mathrm { e } } 3 \right)$ is equal to $\_\_\_\_$

Q89. Let $y = y ( x )$ be the solution of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } y = x \mathrm { e } ^ { \frac { 1 } { \left( 1 + x ^ { 2 } \right) } } ; y ( 0 ) = 0$. Then the area enclosed by the curve $f ( x ) = y ( x ) \mathrm { e } ^ { - \frac { 1 } { \left( 1 + x ^ { 2 } \right) } }$ and the line $y - x = 4$ is $\_\_\_\_$

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.

Let $t$ be a real independent variable, and let $a ( t ) , x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Let $a ( t )$ be a continuous and periodic function of $t$ whose period is $T$. Find the initial-value-problem solution $x ( t )$ of the ordinary differential equation
$$\frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) = a ( t ) x ( t ) ,$$
where $x ( 0 ) = x _ { 0 } \neq 0$.
(2) Find the necessary and sufficient condition on $a ( t )$ such that the solution $x ( t )$ in Question (1) is a periodic solution with a period $T$.
(3) Find the initial-value-problem solutions $x ( t )$ and $y ( t )$ of the simultaneous ordinary differential equations
$$\begin{aligned} \frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) & = - k x ( t ) + \sin ( t ) \cos ( t ) y ( t ) \\ \frac { \mathrm { d } } { \mathrm {~d} t } y ( t ) & = ( - k + \sin ( t ) ) y ( t ) \end{aligned}$$
where $k$ is a real constant, $x ( 0 ) = x _ { 0 } \neq 0$ and $y ( 0 ) = y _ { 0 } \neq 0$.
(4) Explain briefly how $x ( t )$ and $y ( t )$ converge as $t \rightarrow \infty$ when $k > 0$ in Question (3).
(5) Draw a schematic graph of the solution trajectory in Question (3) on the $y x$-plane where $k = 0 , x _ { 0 } = 2$ and $y _ { 0 } = 1$.