If $\frac{d^2y}{dx^2} = 0$,then:

The order of the differential equation whose general solution is given by $y = (C_1 + C_2) \sin (x + C_3) - C_4 e^{x + C_5}$ is (where $C_1, C_2, C_3, C_4, C_5$ are arbitrary constants).

If the family of curves $y = a e^{4x} + b e^{-x}$,where $a, b$ are arbitrary constants,represents the general solution of the differential equation $f(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}) = 0$,then find $\frac{df}{dx}$.

$y = \frac{x}{x + 1}$ is a solution of which differential equation?

Verify that the given function $x+y=\tan ^{-1} y$ is a solution of the differential equation $y^{2} y^{\prime}+y^{2}+1=0$.

The differential equation corresponding to the family of circles in the plane touching the $Y$-axis at the origin is: