Final answer:
The solution to the differential equation involves finding the homogeneous and particular solutions and applying the initial conditions. The beats phenomenon occurs due to the near resonance condition of the driving force frequency compared to the natural frequency of the system, resulting in an interference pattern in amplitude over time.
Step-by-step explanation:
To solve the initial value problem x" + 23.04x = 3cos(5t), x(0) = x'(0) = 0, we are dealing with a forced harmonic oscillator equation. The general solution to this differential equation is a combination of the homogeneous solution (related to the natural motion of the system without external forces) and the particular solution (related to the forced motion due to the external driving force).
The natural frequency of the oscillator is √23.04 rad/s. Since the driving force has a frequency of 5 rad/s, which is close but not equal to the natural frequency, the system undergoes beats, which is an interference pattern of amplitude resulting from two nearby frequencies.
To visualize this, we can use software to graph the function over time. The phenomenon of beats occurs precisely because of the difference in frequencies. If we denote the frequency of the solution as f and the frequency of the driving force as F, the length of each beat will be given by the formula T = 1/|f - F|.
The solution to this initial value problem can be found by combining the homogeneous solution to the associated homogenous equation x" + 23.04x = 0 and a particular solution to the inhomogeneous equation. Because the system starts from rest, we must apply the initial conditions to the complete solution to determine the constants associated with the homogeneous part.