where,
m is the mass of the system,
k is the stiffness of the system,
x is the displacement of the system and ,
is the acceleration.

Solving as a initial value problem, the boundary conditions are,

he initial displacement and velocity is given as,

Where,

is the displacement at time (t) =0 and is the velocity at time (t)=0.

Substituting in equation (1), x(t) = ,we get,

+ k = 0