Final answer:
The solution to the given initial value problem involving undamped forced oscillation will demonstrate pure resonance as the driving force's frequency matches the natural frequency of the system. The associated homogeneous solution and a particular solution that reflects the growing amplitude must be determined to satisfy the initial conditions.
Step-by-step explanation:
The student asked to find the solution of the initial value problem x" + 16x = 24 sin(4t), with the initial conditions x(0) = x'(0) = 0, and then graph the solution to confirm the phenomenon of Pure Resonance.
In forced oscillations, resonance occurs when the driving force's frequency matches the system's natural frequency. Here, the differential equation represents an undamped forced oscillation with a driving force of 24 sin(4t).
To solve this, we use the method of undetermined coefficients to guess a particular solution that resembles the driving force. Since the natural frequency of the system is 4 rad/s (as √16 = 4) and the driving force's frequency is also 4 rad/s, pure resonance occurs leading to a solution with an amplitude that increases over time. The general solution will have the form A cos(4t) + B sin(4t), where A and B are determined by initial conditions, along with a particular solution that accounts for the growing amplitude due to resonance.