Question

This is a differential equation of 2nd order, I need to find a function that if substituted on both sides will be equal

Answer 1

\[\frac{ d ^{2} x }{ dt ^{2} }= -\omega ^{2}x\]

Answer 2

You have constant coefficients in the homogeneous equation,
\[\frac{d^2x}{dt^2}+\omega^2x=0\]
which has the characteristic equation,
\[r^2+\omega^2=0\]
with roots \(r=\pm\sqrt{-\omega}=\pm i\omega\).

The general solution would then be
\[Ce^{\large i\omega t}=C_1\cos\omega t+C_2\sin\omega t\]