# 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 with respect to $x$:

 $\ds 2 y \dfrac {\d y} {\d x}$ $=$ $\ds 4 a$ Power Rule for Derivatives $\ds \leadsto \ \$ $\ds y \dfrac {\d y} {\d x}$ $=$ $\ds 2 a$ $\ds \leadsto \ \$ $\ds y \dfrac {\d^2 y} {\d x^2} + \paren {\dfrac {\d y} {\d x} }^2$ $=$ $\ds 0$ Product Rule for Derivatives

$\blacksquare$