The solution of the differential equation $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ is

The differential equation $\frac{dx}{dy} = \frac{3y}{2x}$ represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity:

The solution of the differential equation $\cos^2 x \frac{d^2y}{dx^2} = 1$ is

The solution of $y{e^{ - x/y}}dx - (x{e^{ - x/y}} + {y^3})dy = 0$ 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?
$(A) P = y + x$
$(B) P = y - x$
$(C) P + Q = 1 - x + y + y^{\prime} + (y^{\prime})^2$
$(D) P - Q = x + y - y^{\prime} - (y^{\prime})^2$

Match the statements/expressions given in Column $I$ with the values given in Column $II$.

Column $I$	Column $II$
$(A)$ The number of solutions of the equation $x e^{\sin x}-\cos x=0$ in the interval $(0, \frac{\pi}{2})$	$(p)$ $1$
$(B)$ Value$(s)$ of $k$ for which the planes $k x+4 y+z=0, 4 x+k y+2 z=0$ and $2 x+2 y+z=0$ intersect in a straight line	$(q)$ $2$
$(C)$ Value$(s)$ of $k$ for which $|x-1|+|x-2|+|x+1|+|x+2|=4 k$ has integer solution$(s)$	$(r)$ $3$
$(D)$ If $y^{\prime}=y+1$ and $y(0)=1$,then value$(s)$ of $y(\ln 2)$	$(s)$ $4$
	$(t)$ $5$