Problem 06 | Elimination of Arbitrary Constants

$x = c_1 \cos \omega t + c_2 \sin \omega t$   ←   equation (1)

$\dfrac{dx}{dt} = -\omega c_1 \, \sin \omega t + \omega c_2 \, \cos \omega t$

$\dfrac{d^2x}{dt^2} = -\omega^2 c_1 \, \cos \omega t - \omega^2 c_2 \, \sin \omega t$

$\dfrac{d^2x}{dt^2} = -\omega^2 (c_1 \, \cos \omega t + c_2 \, \sin \omega t)$
 

From equation (1), c₁ cos ωt + c₂ sin ωt = x
$\dfrac{d^2x}{dt^2} = -\omega^2 x$

$\dfrac{d^2x}{dt^2} + \omega^2 x = 0$           answer