Is the Homogeneous Solution Neglected in Undamped Harmonic Oscillators?

[LydiaAC]

Hello friends:
I do not understand why when solving the undamped harmonic oscillator equation
dx/dt+w02x=Fcoswt I am allowed to neglect the homogeneous solution.
I read that in a damped harmonic oscillator if you let the time pass, the homogeneous solution will disappear and you will keep only the particular solution. But here is no damping!
What is exactly the mathematical justification to set both constants in the homogeneous solution to zero? Or is it an acknowledgment that undamped oscillators do not exist in the physical world and we think of them as having a differential damping, so in an infinite time, we will have the same that in a damped oscillator?
I am very confused.
Lydia

 

[HallsofIvy]

Hello friends:
I do not understand why when solving the undamped harmonic oscillator equation
dx/dt+w02x=Fcoswt I am allowed to neglect the homogeneous solution.
I read that in a damped harmonic oscillator if you let the time pass, the homogeneous solution will disappear and you will keep only the particular solution. But here is no damping!
What is exactly the mathematical justification to set both constants in the homogeneous solution to zero? Or is it an acknowledgment that undamped oscillators do not exist in the physical world and we think of them as having a differential damping, so in an infinite time, we will have the same that in a damped oscillator?
I am very confused.
Lydia

I'm afraid you are going to have to state exactly why you were told that you are "allowed to ignore the homogenous solution" and what the problem is that you are talking about. There is no justification for setting the coefficients in the homogeneous solution equal to 0 unless there are initial conditions that make that appropriate.

 

[LydiaAC]

Hello:
My book on Solid State and Semiconductor Physics (J.P. McKelvey) only assume the particular solution without explanation. I tried to solve equation in a logical way and I find an homogeneous solution plus a particular solution.
Asenw0t+Bcosw0t+F*w0/(w0^2-w^2)coswt
I thought that maybe the book omitted the boundary conditions and I tried to find which of them were used. By assuming that initial velocity is zero you can set A=0 without problem. But the only way to set B=0 is set initial displacement as F*w0/(w0^2-w^2) and I don´t have a reason to do that.
I found in internet this link
http://richter.uprm.edu/~jclinton/Lectures/3/3sdof.htm
I read the following there:
"Since the cosine function is truly periodic and has no beginning or end, there are no initial conditions or transient solutions to deal with. That is, the solution consists of just the particular solution. As it turns out, when a linear system is harmonically forced at one frequency, then the resulting motions (except for transients) are also harmonic at that frequency"...

Later, when solving the problem with other forcing function, it says:

"Unlike the problem of the harmonically driven oscillator, for which the solution was entirely the particular solution, the complementary solution to the homogeneous equation (the transient solution) is very important"

Thank you for your help
Lydia

 

[CPL.Luke]

essentially they are saying that the solutions will oscillate with frequency w rather than w_0 (as a general tendency, and as such the only solution that mattes is theparticular solution

 

[LydiaAC]

Why?
If I had a physical system which have very little damping as a superconductor with resistance zero, didn´t I have two frequencies, the natural and the forced?
When there is damping, they do take account of two frequencies, Isn´t the undamped case a special case of the damped case or there is a discontinuity in behaviour?
Lydia