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) $\frac { 3 } { 1 - ( 8 ) ^ { \frac { 1 } { 4 } } }$

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 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 $\_\_\_\_$

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 $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ 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 $x = x ( y )$ 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 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