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