Let $y = y ( x )$ satisfies the equation $\frac { d y } { d x } - | A | = 0$, for all $x > 0$, where $A = \left[ \begin{array} { c c c } y & \sin x & 1 \\ 0 & - 1 & 1 \\ 2 & 0 & \frac { 1 } { x } \end{array} \right]$. If $y ( \pi ) = \pi + 2$, then the value of $y \left( \frac { \pi } { 2 } \right)$ is:
(1) $\frac { \pi } { 2 } + \frac { 4 } { \pi }$
(2) $\frac { \pi } { 2 } - \frac { 1 } { \pi }$
(3) $\frac { 3 \pi } { 2 } - \frac { 1 } { \pi }$
(4) $\frac { \pi } { 2 } - \frac { 4 } { \pi }$

If the solution curve of the differential equation $\left( 2 x - 10 y ^ { 3 } \right) d y + y d x = 0$, passes through the points $( 0,1 )$ and $( 2 , \beta )$, then $\beta$ is a root of the equation? (1) $y ^ { 5 } - 2 y - 2 = 0$ (2) $y ^ { 5 } - y ^ { 2 } - 1 = 0$ (3) $2 y ^ { 5 } - y ^ { 2 } - 2 = 0$ (4) $2 y ^ { 5 } - 2 y - 1 = 0$

If $y = y ( x )$ is the solution of the differential equation $x \frac { d y } { d x } + 2 y = x e ^ { x } , y ( 1 ) = 0$ then the local maximum value of the function $z ( x ) = x ^ { 2 } y ( x ) - e ^ { x } , x \in R$ is
(1) $1 - e$
(2) 0
(3) $\frac { 1 } { 2 }$
(4) $\frac { 4 } { e } - e$

If $\frac { d y } { d x } + e ^ { x } \left( x ^ { 2 } - 2 \right) y = \left( x ^ { 2 } - 2 x \right) \left( x ^ { 2 } - 2 \right) e ^ { 2 x }$ and $y ( 0 ) = 0$, then the value of $y ( 2 )$ is
(1) $-1$
(2) 1
(3) 0
(4) $e$

Let the solution curve $y = y ( x )$ of the differential equation $\left( 4 + x ^ { 2 } \right) dy - 2 x \left( x ^ { 2 } + 3 y + 4 \right) dx = 0$ pass through the origin. Then $y ( 2 )$ is equal to $\_\_\_\_$.

Let $y = y ( x )$ be the solution of the differential equation $\left( 1 - x ^ { 2 } \right) d y = \left( x y + \left( x ^ { 3 } + 2 \right) \sqrt { 1 - x ^ { 2 } } \right) d x , - 1 < x < 1$ and $y ( 0 ) = 0$. If $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sqrt { 1 - x ^ { 2 } } y ( x ) d x = k$ then $k ^ { - 1 }$ is equal to

The general solution of the differential equation $(x - y^2)dx + y(5x + y^2)dy = 0$ is
(1) $y ^ { 2 } + x ^ { 4 } = C(y ^ { 2 } + 2x ^ { 3 })$
(2) $y ^ { 2 } + 2x ^ { 4 } = C(y ^ { 2 } + x ^ { 3 })$
(3) $y ^ { 2 } + x ^ { 3 } = C(2y ^ { 2 } + x ^ { 4 })$
(4) $y ^ { 2 } + 2x ^ { 3 } = C(2y ^ { 2 } + x ^ { 4 })$

Let $y = y ( x )$ be the solution curve of the differential equation $\sin \left( 2 x ^ { 2 } \right) \log _ { e } \left( \tan x ^ { 2 } \right) d y + \left( 4 x y - 4 \sqrt { 2 } x \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) \right) d x = 0,0 < x < \sqrt { \frac { \pi } { 2 } }$, which passes through the point $\left( \sqrt { \frac { \pi } { 6 } } , 1 \right)$. Then $\left| y \left( \sqrt { \frac { \pi } { 3 } } \right) \right|$ is equal to $\_\_\_\_$ .

Let $y = y ( x )$ be the solution curve of the differential equation $\frac { d y } { d x } + \frac { 1 } { x ^ { 2 } - 1 } y = \left( \frac { x - 1 } { x + 1 } \right) ^ { \frac { 1 } { 2 } } , x > 1$ passing through the point $\left( 2 , \sqrt { \frac { 1 } { 3 } } \right)$. Then $\sqrt { 7 } y ( 8 )$ is equal to
(1) $11 + 6 \log _ { e } 3$
(2) $19$
(3) $12 - 2 \log _ { e } 3$
(4) $19 - 6 \log _ { e } 3$

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

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

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 $\_\_\_\_$

If $y = y(x)$ and $\left(1 + x^{2}\right)dy + \left(1 - \tan^{-1}x\right)dx = 0$ and $y(0) = 1$ then $\mathbf{y}(1)$ is
(A) $\frac{\pi^{2}}{32} + \frac{\pi}{4} + 1$ (B) $\frac{\pi^{2}}{32} - \frac{\pi}{2} + 1$ (C) $\frac{\pi^{2}}{32} + \frac{\pi}{2} - 1$ (D) $\frac{\pi^{2}}{32} - \frac{\pi}{4} + 1$

Consider the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$
If $y(0) = -\frac{7}{4}$, then find $y\left(\frac{\pi}{6}\right)$.
(A) $-5/2$
(B) $0$
(C) $1$
(D) $3/2$

The solution of the differential equation $\mathbf { x d y } - \mathbf { y d x } = \sqrt { \mathbf { x } ^ { 2 } + \mathbf { y } ^ { 2 } } \mathbf { d x }$ is (where c is integration constant)
(A) $\sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } = \mathrm { cx } ^ { 2 } - \mathrm { y }$
(B) $\sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } = \mathrm { cx } ^ { 2 } + \mathrm { y }$
(C) $\sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } = \mathrm { cx } - \mathrm { y }$
(D) $\sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } = \mathrm { cx } + \mathrm { y }$