The order of the differential equation of all circles whose radius is $4$,is . . . . . .

The differential equation representing the family of curves $x \sin x + y^3 = 4ax$ is

The differential equation representing the family of circles having their centres on the $Y$-axis is (where $y_1 = \frac{dy}{dx}$ and $y_2 = \frac{d^2y}{dx^2}$):

If $c$ is a parameter,then the differential equation of the family of curves $x^2=c(y+c)^2$ is

The differential equation of the family of circles whose center lies on the $x$-axis is:

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 $\frac{df}{dx}$.