Theelectricchild
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I am seeking information on the following problem:
The problem statement tells that a function z(t) = x(t) + j y(t) (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)
z'' + 2bz' + \omega^2 z = Fe^{j\omega_ot}
All derivatives are with respect to time, and j = root -1.
And the question states to find the DIFF EQN that is satisfied by x(t) that is the real part of z(t).
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 don't 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.
The problem statement tells that a function z(t) = x(t) + j y(t) (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)
z'' + 2bz' + \omega^2 z = Fe^{j\omega_ot}
All derivatives are with respect to time, and j = root -1.
And the question states to find the DIFF EQN that is satisfied by x(t) that is the real part of z(t).
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 don't 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.