The solution curve of the differential equation $y \frac { d x } { d y } = x \left( \log _ { e } x - \log _ { e } y + 1 \right) , \quad x > 0 , \quad y > 0$ passing through the point $(e, 1)$ is
(1) $\log _ { e } \frac { y } { x } = x$
(2) $\log _ { e } \frac { y } { x } = y ^ { 2 }$
(3) $\log _ { e } \frac { x } { y } = y$
(4) $2 \log _ { e } \frac { x } { y } = y + 1$

Let $f ( x )$ be a positive function such that the area bounded by $y = f ( x ) , y = 0$ from $x = 0$ to $x = a > 0$ is $e ^ { - a } + 4 a ^ { 2 } + a - 1$. Then the differential equation, whose general solution is $y = c _ { 1 } f ( x ) + c _ { 2 }$, where $c _ { 1 }$ and $c _ { 2 }$ are arbitrary constants, is
(1) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$
(2) $\left( 8 e ^ { x } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(3) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - \frac { d y } { d x } = 0$
(4) $\left( 8 e ^ { x } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } = 0$

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$ and $\lim_{x \to -\infty} f(x) = 1$. If $f'(x) = \alpha f(x) + 3$, for all $x \in \mathbb{R}$, then $f(-\log_e 2)$ is equal to:
(1) 1
(2) 5
(3) 9
(4) 7

If $\sin \left( \frac { y } { x } \right) = \log _ { e } | x | + \frac { \alpha } { 2 }$ is the solution of the differential equation $x \cos \left( \frac { y } { x } \right) \frac { d y } { d x } = y \cos \left( \frac { y } { x } \right) + x$ and $y ( 1 ) = \frac { \pi } { 3 }$, then $\alpha ^ { 2 }$ is equal to
(1) 3
(2) 12
(3) 4
(4) 9

A function $y = f ( x )$ satisfies $f ( x ) \sin 2 x + \sin x - \left( 1 + \cos ^ { 2 } x \right) f ^ { \prime } ( x ) = 0$ with condition $f ( 0 ) = 0$. Then $f \left( \frac { \pi } { 2 } \right)$ is equal to
(1) 1
(2) 0
(3) - 1
(4) 2

If $y = y ( x )$ is the solution curve of the differential equation $x ^ { 2 } - 4 d y - y ^ { 2 } - 3 y d x = 0 , x > 2 , y ( 4 ) = \frac { 3 } { 2 }$ and the slope of the curve is never zero, then the value of $\mathrm { y } ( 10 )$ equals :
(1) $\frac { 3 } { 1 + ( 8 ) ^ { \frac { 1 } { 4 } } }$
(2) $\frac { 3 } { 1 + 2 \sqrt { z } }$
(3) $\frac { 3 } { 1 - 2 \sqrt { 2 } }$
(4) $\frac { 3 } { 1 - ( 8 ) ^ { \frac { 1 } { 4 } } }$

The differential equation of the family of circles passing through the origin and having centre at the line $y = x$ is :
(1) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(2) $\left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } y$
(3) $\left( x ^ { 2 } + y ^ { 2 } - 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } + y ^ { 2 } + 2 x y \right) \mathrm { d } y$
(4) $\left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } x = \left( x ^ { 2 } - y ^ { 2 } + 2 x y \right) \mathrm { d } y$

Suppose the solution of the differential equation $\frac { d y } { d x } = \frac { ( 2 + \alpha ) x - \beta y + 2 } { \beta x - 2 \alpha y - ( \beta \gamma - 4 \alpha ) }$ represents a circle passing through origin. Then the radius of this circle is:
(1) 2
(2) $\sqrt { 17 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { \sqrt { 17 } } { 2 }$

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) $\frac { \pi } { 16 }$

If the solution of the differential equation $( 2 x + 3 y - 2 ) d x + ( 4 x + 6 y - 7 ) d y = 0 , y ( 0 ) = 3$, is $\alpha x + \beta y + 3 \log _ { e } | 2 x + 3 y - \gamma | = 6$, then $\alpha + 2 \beta + 3 \gamma$ is equal to $\_\_\_\_$.

Let $y = y ( x )$ be the solution of the differential equation $( x + y + 2 ) ^ { 2 } d x = d y , y ( 0 ) = - 2$. Let the maximum and minimum values of the function $y = y ( x )$ in $\left[ 0 , \frac { \pi } { 3 } \right]$ be $\alpha$ and $\beta$, respectively. If $( 3 \alpha + \pi ) ^ { 2 } + \beta ^ { 2 } = \gamma + \delta \sqrt { 3 } , \gamma , \delta \in \mathbb { Z }$, then $\gamma + \delta$ equals $\_\_\_\_$

If the solution curve, of the differential equation $\frac { d y } { d x } = \frac { x + y - 2 } { x - y }$ passing through the point $( 2,1 )$ is $\tan ^ { - 1 } \frac { y - 1 } { x - 1 } - \frac { 1 } { \beta } \log _ { e } \alpha + \frac { y - 1 } { x - 1 } ^ { 2 } = \log _ { e } x - 1$, then $5 \beta + \alpha$ is equal to $\_\_\_\_$.

Let $Y = Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y - y = Y'(x)(X - x)$ and the coordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)} + 1$, $Y'(x) \neq 0$. If $Y(1) = 1$, then $12\,Y(2)$ equals $\underline{\hspace{1cm}}$.

Let $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ be the solution of the differential equation $x \mathrm {~d} y - y \mathrm {~d} x + x y ( x \mathrm {~d} y + y \mathrm {~d} x ) = 0$, $y ( 1 ) = 2$. Then $\alpha + \beta$ is equal to $\_\_\_\_$

Let a curve $y = f ( x )$ pass through the points $( 0,5 )$ and $\left( \log _ { e } 2 , k \right)$. If the curve satisfies the differential equation $2 ( 3 + y ) e ^ { 2 x } d x - \left( 7 + e ^ { 2 x } \right) d y = 0$, then $k$ is equal to
(1) 4
(2) 32
(3) 8
(4) 16

Let $f : (0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to:
(1) 39
(2) 19
(3) 29
(4) 23

Let $f ( x )$ be a real differentiable function such that $f ( 0 ) = 1$ and $f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y )$ for all $x , y \in \mathbf { R }$. Then $\sum _ { \mathrm { n } = 1 } ^ { 100 } \log _ { \mathrm { e } } f ( \mathrm { n } )$ is equal to:
(1) 2525
(2) 5220
(3) 2384
(4) 2406

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

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1+x^2}\right)dx + \left(1+x^2\right)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to
(1) $\sqrt{\frac{15}{2}}$
(2) $\frac{5\sqrt{3}}{2}$
(3) $2\sqrt{2}$
(4) $\sqrt{\frac{14}{3}}$

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 $\_\_\_\_$

Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt$, $x \geq 0$. Then $f(2)$ is equal to $\underline{\hspace{2cm}}$.

For each of Q , S , V in the following sentences, choose the appropriate expression from among (0) $\sim$ (7) at the bottom of this page. For the other $\square$, enter the correct number.
Suppose we have a differentiable function $f ( x )$ which satisfies the equation
$$\int _ { 0 } ^ { x } f ( t ) d t = \left( 1 + e ^ { - x } \right) f ( x ) + 2 x - 4 \log 2 \tag{1}$$
We are to find $f ( x )$ and the value of $\lim _ { x \rightarrow \infty } f ( x )$.
When we differentiate each side of (1) with respect to $x$ and transform the equation, we have
$$\left( 1 + e ^ { - x } \right) ( \mathbf { Q } ) = \mathbf { R } . \tag{2}$$
Next we set $f ( x ) = e ^ { x } g ( x )$, and using (2), we obtain
$$g ^ { \prime } ( x ) = \frac { \mathbf { S } } { 1 + e ^ { - x } }$$
and hence
$$g ( x ) = \mathbf { T } \log \left( 1 + e ^ { - x } \right) + C ,$$
where $C$ is an integral constant. Furthermore, since $g ( 0 ) = f ( 0 )$, we see that $C = \mathbf { U }$. Thus we obtain $g ( x )$ and from that,
$$f ( x ) = \mathbf { V } \log \left( 1 + e ^ { - x } \right) .$$
Finally, we set $e ^ { - x } = t$ and obtain
$$f ( x ) = \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } }$$
and hence
$$\lim _ { x \rightarrow \infty } f ( x ) = \lim _ { t \rightarrow \mathbf { X } } \mathbf { W } \log ( 1 + t ) ^ { \frac { 1 } { t } } = \mathbf { Y }$$
Choices: (0) $f ^ { \prime } ( x ) - f ( x )$
(1) $f ( x ) - f ^ { \prime } ( x )$
(2) $f ^ { \prime } ( x ) - 2 f ( x )$
(3) $f ( x ) - 2 f ^ { \prime } ( x )$
(4) $2 e ^ { x }$
(5) $- 2 e ^ { x }$ (6) $2 e ^ { - x }$ (7) $- 2 e ^ { - x }$

The Laplace transform of the function $f ( t )$, defined for $t \geq 0$, is denoted by $F ( s ) = \mathcal { L } [ f ( t ) ]$ and its definition is given by
$$F ( s ) = \mathcal { L } [ f ( t ) ] = \int _ { 0 } ^ { \infty } f ( t ) \exp ( - s t ) d t$$
where $s$ is a complex number. In the following, the set of all complex numbers is denoted by $\mathbb { C }$, and the set of the complex numbers with positive real parts is denoted by $\mathbb { C } ^ { + }$.
I. Consider the following function $g ( t )$ defined for $t \geq 0$ :
$$g ( t ) = \int _ { 0 } ^ { \infty } \frac { \sin ^ { 2 } ( t x ) } { x ^ { 2 } } d x$$

1. Find the Laplace transform $G ( s ) = \mathcal { L } [ g ( t ) ] \left( s \in \mathbb { C } ^ { + } \right)$ of the function $g ( t )$.
2. Obtain the value of the following integral using the result of Question I.1: $$\int _ { - \infty } ^ { \infty } \frac { \sin ^ { 2 } ( x ) } { x ^ { 2 } } d x$$

II. Consider the function $u ( x , t )$ that satisfies the following partial differential equation:
$$\frac { \partial u ( x , t ) } { \partial t } = \frac { \partial ^ { 2 } u ( x , t ) } { \partial x ^ { 2 } } ( 0 < x < 1 , t > 0 )$$
under the boundary conditions:
$$\left\{ \begin{array} { l } \left. \frac { \partial u ( x , t ) } { \partial x } \right| _ { x = 0 } = 0 \quad ( t \geq 0 ) \\ u ( 1 , t ) = 1 \quad ( t \geq 0 ) \\ u ( x , 0 ) = \frac { \cosh ( x ) } { \cosh ( 1 ) } \quad ( 0 < x < 1 ) \end{array} \right.$$

1. The Laplace transform of $u ( x , t )$ is denoted by $U ( x , s ) = \mathcal { L } [ u ( x , t ) ]$ ( $s \in \mathbb { C } ^ { + }$). Derive the ordinary differential equation and boundary conditions for $U ( x , s )$ with respect to the independent variable $x$. Here, the function $u ( x , t )$ can be assumed to be bounded. The following relations can also be used: $$\begin{aligned} & \mathcal { L } \left[ \frac { \partial u ( x , t ) } { \partial x } \right] = \frac { \partial U ( x , s ) } { \partial x } \\ & \mathcal { L } \left[ \frac { \partial ^ { 2 } u ( x , t ) } { \partial x ^ { 2 } } \right] = \frac { \partial ^ { 2 } U ( x , s ) } { \partial x ^ { 2 } } \end{aligned}$$
2. Using an analytic function $Q ( s ) ( s \in \mathbb { C } )$, the function $U _ { \mathrm { c } } ( x , s )$ is defined as follows: $$U _ { c } ( x , s ) = \frac { \cosh ( x ) } { ( s - 1 ) \cosh ( 1 ) } - \frac { \cosh ( x \sqrt { s } ) } { Q ( s ) } \quad ( 0 \leq x \leq 1 )$$ When the function $U ( x , s ) = U _ { \mathrm { c } } ( x , s )$ satisfies the differential equation and the boundary conditions derived in Question II.1 for $s \in \mathbb { C } ^ { + }$, find the function $Q ( s )$.
3. Using the function $Q ( s )$ derived in Question II.2, the sequence of complex numbers $\left\{ a _ { r } \right\} ( r = 1,2 , \cdots )$ is defined by arranging all of the roots of $Q ( s ) = 0 ( s \in \mathbb { C } )$ in ascending order of their absolute values. In this case, the following limits $R _ { r } ( x , t )$ are finite for $t \geq 0,0 \leq x \leq 1$, and $r \geq 1$ : $$R _ { r } ( x , t ) = \lim _ { s \rightarrow a _ { r } } \left( s - a _ { r } \right) U _ { \mathrm { c } } ( x , s ) \exp ( s t )$$ and the solution of the partial differential equation is given by $$u ( x , t ) = \sum _ { r = 1 } ^ { \infty } R _ { r } ( x , t )$$ Determine $R _ { 1 } ( x , t ) , R _ { 2 } ( x , t )$, and $R _ { r } ( x , t )$ for $r \geq 3$.

$\frac{1}{x + 1} + x - 1 = \frac{1}{x^{2}}$
Given that, which of the following is the expression $x^{3} - 1$ equal to?
A) $\frac{2}{x - 1}$ B) $\frac{1}{x}$ C) $\frac{x - 1}{x}$ D) $-x$ E) $\frac{1}{x + 1}$

A function f defined on the set of natural numbers is defined for every n as
$$f ( n ) = \begin{cases} 5 n + 40 , & 0 \leq n < 10 \\ f ( n - 10 ) , & n \geq 10 \end{cases}$$
Example: $f ( 23 ) = f ( 13 ) = f ( 3 ) = 5 \cdot 3 + 40 = 55$
Accordingly, what is the sum of the two-digit numbers AB that satisfy the equation $f ( A B ) = A B$?
A) 75 B) 80 C) 90 D) 100 E) 105