A weight of 3.2Ib stretches a spring 6.4 ft. If the weight is released 3 inches below the equilibrium position with a downward directed speed of 6 ft/s and the mass-spring system is in the middle provides a damping force numerically equal to one fifth of the instantaneous speed, determine the equation of motion if an external force given by  f(t) = e-t cos(2t) is also applied to weight.

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.