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\frac{d^2 y}{dx^2} = y" = - \omega^2\ y \\

This is the equation of motion of S H M or simple harmonic motion. It is applicable for oscillations of a mass m attached at the end of a spring whose other end is tied to a wall or a rigid body.   The same equation is applicable in a circular motion also.

Let y = A sin (ωx +Ф)
y' = d y/ d x  = Aω cos (ωt + Ф) 
y" = d² y / d x² = - A ω² sin (ω t + Ф ) = - A ω² y

So A = 1 and the solution is y = Sin (ω t + Ф)

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