Reference: "An Introduction to Mechanics" by D. Kleppner and R. J. Kolenkow, 2nd Edition

Note: This problem does not come from the textbook, as it is from class homework. Please note that the textbook is not necessary to solve this problem; all information needed is included in the problem.

Consider an overdamped oscillator having the form:

x(t)+x(t)+02x(t)=0

The general solution is x(t) = Ae^i₊t + Be^i_-t, where ₊ and _- are imaginary, and A and B are constants determined by the initial conditions. Find A and B in terms of the initial position x(0) and the initial velocity v(0). Show that in the limit of very large , A is determined primarily by v(0) and B is determined primarily by x(0). Additionally, show that the overdamped oscillator crosses the equilibrium point (i.e., x = 0) at most once.