Formation of Ordinary Differential Equation by Elimination/Examples/Simple Harmonic Motion
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Examples of Formation of Ordinary Differential Equation by Elimination
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 $3$:
- $y \cdot \dfrac {\d^3 y} {\d x^3} = \dfrac {\d y} {\d x} \cdot \dfrac {\d^2 y} {\d x^2}$
Proof
This equation has a dependent variable of $y$ and an independent variable of $x$.
It is required to eliminate the arbitrary constants $A$, $\phi$ and $\omega$.
Differentiating $2$ times with respect to $x$:
\(\text {(2)}: \quad\) | \(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds -\omega A \map \sin {\omega x + \phi}\) | Derivative of Cosine Function | ||||||||||
\(\text {(3)}: \quad\) | \(\ds \dfrac {\d^2 y} {\d x^2}\) | \(=\) | \(\ds -\omega^2 A \map \cos {\omega x + \phi}\) | Derivative of Sine Function | ||||||||||
\(\ds \) | \(=\) | \(\ds -\omega^2 y\) |
We have arrived at a ordinary differential equation of order $2$:
- $\dfrac {\d^2 y} {\d x^2} = -\omega^2 y$
Now we differentiate one further time with respect to $x$:
\(\ds \map {\dfrac \d {\d x} } {\dfrac {\d^2 y} {\d x^2} }\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {-\omega^2 y}\) | ||||||||||||
\(\text {(4)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \dfrac {\d^3 y} {\d x^3}\) | \(=\) | \(\ds -\omega^2 \dfrac {\d y} {\d x}\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d^3 y} {\d x^3} / \dfrac {\d y} {\d x}\) | \(=\) | \(\ds -\omega^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\d^2 y} {\d x^2} / y\) | from $(3)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \cdot \dfrac {\d^3 y} {\d x^3}\) | \(=\) | \(\ds \dfrac {\d y} {\d x} \cdot \dfrac {\d^2 y} {\d x^2}\) | multiplying up |
$\blacksquare$
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