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

$\blacksquare$

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