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The problem statement tells that a function [tex]z(t) = x(t) + j y(t)[/tex] (where x(t) and y(t) are real functions of t) satisfies the following Inhomogenous differential equation (That i can see corresponds to a damped, driven harmonic oscillator)

[tex]z'' + 2bz' + \omega^2 z = Fe^{j\omega_ot}[/tex]

All derivatives are with respect to time, and j = root -1.

And the question states to find the DIFF EQN that is satisfied by [tex]x(t)[/tex] that is the real part of [tex]z(t)[/tex].

At first, I was thinking, since x(t) corresponds to the real part of the solution to the DE above, that the differential equation must not have a characteristic equation with complex solutions ( that is [b^2 - 4w^2] CANNOT be less than zero, and therefore b^2 = 4w^2 AT LEAST. But I am given no value of any of the parameters in the DE. I also thought about excluding the right side forcing function also because that has complex numbers in it, so i assumed that we cannot include that in the answer--- however I am not sure. I am not being asked to solve the DE above, but rather find one for x(t), but I tried solving the DE anyway to see what the COMPLEX part and the REAL part would be and perhaps reverse engineer the problem back to get a DE that would be the answer... the way to do that would be to either use a particular solution, laplace transform or variation of parameters... but I dont want to waste my time doing that since we are "technically" not supposed to know how to solve Inhomog DEs just yet.

Any help would be appreciated, thanks a lot.