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 $3$:

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

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

$\blacksquare$