Final answer:
The question deals with finding the particular solution for a driven oscillator equation in Physics, where an oscillating function matching the frequency of the driving force is sought after the transients subside. By comparing the system's characteristics, the damping condition is determined, leading to the solution form for steady state harmonically driven motion.
Step-by-step explanation:
The subject of the question is Physics, specifically focusing on the driven oscillator in harmonic motion described by a differential equation. To solve for the particular solution, we look for a function x(t) that has the same frequency as the driving force but possibly with a phase shift. Since the driving force is given as F0 sin(ωt), the guess for the particular solution will have the form Acos(ωt + φ) or Asin(ωt + φ), where A is the amplitude and φ is the phase shift of the particular solution. By substituting this guess into the differential equation and solving for A and φ considering the specific parameters of the system (m, c, k), we can determine the constants and thereby obtain the particular solution that reaches a steady state after transients die out.
To solve for the particular solution of the given driven oscillator equation mx'' + cx' + kx = F0 sin(ωt), we use an oscillating solution approach assuming a steady state after the transient responses have dissipated. As a function of time, x(t) will have the same frequency as the driving force. The actual form of x(t) and its parameters will depend on the characteristics of the system (mass m, damping coefficient c, spring constant k), and whether it's underdamped, critically damped, or overdamped. Given the provided information, further mathematical operations are required to find the values for the amplitude A and the phase shift φ in terms of system constants and the driving force characteristics.