The differential equation of all parabolas with axis paraIell to the axis of y, is

The equation of the family of parabolas with axis parallel to axis of y is

$(x-a{)}^2=A(y-b)$

This eguation contains three arbitrary constants viz. a, band A. Differentiating with respect to x, we get 

$2(x-a)=A\frac{dy}{dx}$

Differentiating again with respect to x, we get

$2=A\frac{d^{2}y}{d{x}^2}$

Differentiating this with respect to $x$ we get $\frac{d^{3}y}{d{x}^3}=0$