# Question about complex solutions to DiffEqns

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 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.

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Hmmm--- one more thought--- could Euler's Identity $$e^{j \omega t} = cos (\omega t) + j sin(\omega t)$$ help for this problem? I see it could maybe be used to change that forcing function on the right hand side....

Hurkyl
Staff Emeritus
Gold Member
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.
That is incorrect -- complex-linear combinations of your two complex solutions can be purely real.

For example, try solving the ODE $z'' - z = 0$.

Then the method you mentioned will give you these two independent solutions: $e^{jt}$ and $e^{-jt}$. However, as you well know, the solution space is also spanned by these two independent solutions: $\cos t$ and $\sin t$.

Exercise: you've already shown how to write the exponential function as a linear combination of the two trig functions. Now, show how to write the trig functions as linear combinations of the exponential functions.

Anyways, Euler's identity will help -- your goal is to simlpy take the real part of the equation, and that identity tells you the real part of the RHS.

Wait a minute--- What about just having the right hand side be $$Fcos(\omega_o t)$$? The solution to the DE then would be only give a real solution --- wouldnt that correspond to x(t)?

HallsofIvy
Homework Helper
That's pretty much what Hurkyl just said!

dextercioby
Homework Helper
How about pluggin' in the eq.the complex solution and then identifying the real & the imaginary parts of each side of the equation...???

Daniel.

thanks guys i think i have it, and btw i cannot plug in the complex solution because he didnt give me it! He just said it was in x + ji form ---- but your method would have been great had i known them!!

dextercioby
Homework Helper
Well
$$z(t)=:x(t)+jy(t)$$

is the "complex solution" i was referring to...

Daniel...

P.S.YW.

Hurkyl
Staff Emeritus