Question about complex solutions to DiffEqns

[Theelectricchild]

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.

 

[Theelectricchild]

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]

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.

 

[Theelectricchild]

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 --- wouldn't that correspond to x(t)?

 

[HallsofIvy]

That's pretty much what Hurkyl just said!

 

[dextercioby]

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

Daniel.

 

[Theelectricchild]

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]

Well
$$z(t)=:x(t)+jy(t)$$

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

Daniel...


P.S.YW.

 

[Hurkyl]

x(t) + j y(t) is the complex solution.

 

[Theelectricchild]

thanks ill try that.