Which of the following represents the general solution of a damped oscillating system?

In the context of a damped oscillating system, the correct representation of the general solution is a function that describes how the oscillations decrease in amplitude over time due to the presence of damping forces, such as friction or air resistance.

The expression involving the exponential term ( e^{-\frac{bt}{2m}} ) captures this damping effect. The term ( A ) represents the initial amplitude, while the cosine function ( \cos(wt + o) ) indicates a harmonic oscillation, where ( w ) is the angular frequency of oscillation and ( o ) is the phase constant.

As time progresses, the amplitude of the oscillation is reduced by the factor ( e^{-\frac{bt}{2m}} ), showing how the energy in the system dissipates. Thus, this formulation effectively models a damped harmonic oscillator by combining the exponential decay of amplitude due to damping with the oscillatory behavior intrinsic to harmonic motion.

In contrast, the other options do not appropriately represent a damped system. For instance, options with an increasing exponential term, like ( e^{\frac{bt}{2m}} ), suggest an unphysical scenario where the amplitude grows over time. Overall, choice