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

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

The differential equation of all parabolas with axis parallel to the $y$-axis is

For each of the exercises given below,verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
$y=e^{x}(a \cos x+b \sin x) \quad: \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+2 y=0$

The differential equation obtained by eliminating the arbitrary constants from the equation $y^{2}=(2 x+c)^{5}$ is

$A$ differential equation representing the family of parabolas with axis parallel to the $y$-axis and whose length of latus rectum is the distance of the point $(2, -3)$ from the line $3x + 4y = 5$,is given by: