Formation of Ordinary Differential Equation by Elimination/Examples/Parabolas whose Axes are X Axis

Examples of Formation of Ordinary Differential Equation by Elimination
Consider the set of all parabolas embedded in the Cartesian plane whose axis is the $x$ axis.

This set can be expressed as the ordinary differential equation of order $2$:


 * $y \dfrac {\d^2 y} {\d x^2} + \paren {\dfrac {\d y} {\d x} }^2 = 0$

Proof
Such a parabola has the equation:


 * $y^2 = 4 a \paren {x - h}$

Differentiating twice $x$: