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 $t$:

\(\text {(2)}: \quad\) \(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds -\omega A \map \sin {\omega x + \phi}\) 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

$\blacksquare$


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