The major purpose of this project is to describe some special motion in mechanical vibration and electrical RLC circuit problems mathematically by second order ordinary differential equations. Applications of both scalar and systems of second order ordinary differential equation in the analysis of mechanical vibration problem have been demonstrated. The responses of both damped and undamped mass spring systems subjected to linear viscous damping has been analyzed. Application of second order ordinary differential equations in electrical circuit analysis involving resistor, capacitor and inductor has been demonstrated. The responses of both series and parallel RLC circuit subjected to DC and AC sources are analyzed.