In this paper, Application of Second Order ordinary differential equations to Simple harmonic and damped Motion. , simple harmonic system which can easily be solved by period of motion, frequency of motion, equation of motion, amplitude and phase and phase constant. the damping force acts in a direction to the motion and solved by the general solutions of both homogenous(no external force) and non-homogenous (with external force) differential equation within different case and forced motion/driven motion of differential equations. Finally there are four different types of damping which are classified by the outcome of their damping ratio.
γ=0: undamped, γ<1:under damped, γ=1:critically damped, γ>1:over damped
Keywords: simple harmonic motion, damped motion