This is a second-order homogeneous differential equation, and its general solution is

X=Aexp(wt+p), and the coefficient a can be determined by initial conditions.

Because the coefficients are not unique, the special solution of the general solution can be

X=Acos(wt+p), which can also be

X=Asin(wt+p). In college physics, people agree.

X=Acos(wt+p) as the expression of simple harmonic vibration.