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

Jump to navigation
Jump to search

## 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 |

$\blacksquare$

## Sources

- 1952: H.T.H. Piaggio:
*An Elementary Treatise on Differential Equations and their Applications*(revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $5$. Examples