Definition:Formation of Ordinary Differential Equation by Elimination

Definition

Let $\map f {x, y, C_1, C_2, \ldots, C_n} = 0$ be an equation:

whose dependent variable is $y$

whose independent variable is $x$

$C_1, C_2, \ldots, C_n$ are constants which are deemed to be arbitrary.

A differential equation may be formed from $f$ by:

differentiating $n$ times with respect to $x$ to obtain $n$ equations in $x$ and $\dfrac {\d^k y} {\d x^k}$, for $k \in \set {1, 2, \ldots, n}$

eliminating $C_k$ from these $n$ equations, for $k \in \set {1, 2, \ldots, n}$.

Examples

Simple Harmonic Motion

Consider the equation governing simple harmonic motion:

$(1): \quad y = A \map \cos {\omega x + \phi}$

This can be expressed as the ordinary differential equation of order $3$:

$y \cdot \dfrac {\d^3 y} {\d x^3} = \dfrac {\d y} {\d x} \cdot \dfrac {\d^2 y} {\d x^2}$

Parabolas whose Axes are $x$-Axis

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 $2$:

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

Also see

• Results about formation of ordinary differential equations by elimination can be found here.

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: $4$. Formation of differential equations by elimination