Differential Equation governing Simple Harmonic Motion

Theorem

Then its motion can be described using the equation:

$\dfrac {\d^2 x} {\d x^2} = -\omega^2 x$

$(1): \quad x = A \map \cos {\omega t + \phi}$

$\text {(2)}: \quad$

$\ds \dfrac {\d x} {\d t}$

$=$

$\ds -\omega A \map \sin {\omega t + \phi}$

$\text {(3)}: \quad$

$\ds \dfrac {\d^2 x} {\d t^2}$

$=$

$\ds -\omega^2 A \map \cos {\omega t + \phi}$

$\ds $

$=$

$\ds -\omega^2 x$

$\dfrac {\d^2 x} {\d t^2} = -\omega^2 x$

$\blacksquare$