# Definition:Formation of Ordinary Differential Equation by Elimination

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## 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