Relation between a differential equation satisfied by parabolas and a formula for the slope of their tangents

Question

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P.

Statement 2: The system of parabolas $y^2=4ax$ satisfies a differential equation of degree 1 and order 1.

I know and it is easy to prove both (separately) are true, but my question is whether statement 2 is a correct explanation of statement 1?

Answer 1

For $y^2=4ax$

Derivate

$2y\dfrac{dy}{dx}=4a$

$\dfrac{dy}{dx}=\dfrac{4a}{2y}=\dfrac{2a}{y}$

Then

$2adx-ydy=0$