If the curve $y = y(x)$ represented by the solution of the differential equation $(2xy^2 - y)dx + xdy = 0$ passes through the intersection of the lines $2x - 3y = 1$ and $3x + 2y = 8$,then $|y(1)|$ is equal to ...... .

$A$ continuously differentiable function $\phi (x)$ in $(0, \pi)$ satisfying $y' = 1 + y^2$ and $y(0) = 0 = y(\pi)$ is

Solve the differential equation $\left( {1 + x\sqrt {{x^2} + {y^2}} } \right)\,dx + \left( {\sqrt {{x^2} + {y^2}} - 1 } \right)y\,dy = 0$.

Find the family of curves such that the angle between the tangent at any point $(x, y)$ and the tangent to the curve $xy = c^2$ at the point of intersection is $\frac{\pi}{4}$.

The solution of $(xy \cos xy + \sin xy)dx + x^2 \cos xy \, dy = 0$ is

Consider the family of all circles whose centers lie on the straight line $y = x$. If this family of circles is represented by the differential equation $P y^{\prime \prime} + Q y^{\prime} + 1 = 0$,where $P, Q$ are functions of $x, y$ and $y^{\prime}$ (here $y^{\prime} = \frac{dy}{dx}, y^{\prime \prime} = \frac{d^2y}{dx^2}$),then which of the following statements is (are) true?
$(A) P = y + x$
$(B) P = y - x$
$(C) P + Q = 1 - x + y + y^{\prime} + (y^{\prime})^2$
$(D) P - Q = x + y - y^{\prime} - (y^{\prime})^2$