Formation of Ordinary Differential Equation by Elimination/Examples/Simple Harmonic Motion

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 $3$:


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

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 $t$:

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 $x$: