Harmonic oscillator derivation

AI Thread Summary
The discussion centers on deriving the sinusoidal equation for a simple harmonic oscillator, specifically the terms involving P and Q in the equation. Participants clarify that the differential equation governing the harmonic oscillator is linear and homogeneous, leading to a general solution formed by summing solutions from the characteristic equation. A misunderstanding arises from the term "armonic," which is corrected to "harmonic." The importance of understanding ordinary differential equations (ODEs) is emphasized for grasping the derivation. Overall, the conversation highlights the foundational concepts necessary for deriving the equation.
pabilbado
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I was wondering how to derive the sinusoidal equation for the simple armonic oscillator. But I am currently trying to understand this step in this webpage:
f903148331897dd9406ca8fc3e1e150e.png

I don't get where do P and Q come from and why it is summing pe^iwt + qe^-iwt. please I need some help. The rest of it pretty much makes sense.
 
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pabilbado said:
I was wondering how to derive the sinusoidal equation for the simple armonic oscillator. But I am currently trying to understand this step in this webpage:
f903148331897dd9406ca8fc3e1e150e.png

I don't get where do P and Q come from and why it is summing pe^iwt + qe^-iwt. please I need some help. The rest of it pretty much makes sense.
I'm not sure what an "armonic" oscillator is, unless you mean "harmonic".

The differential equation of the harmonic oscillator is linear and homogeneous, so the general solution is composed of the sum of all the solutions obtained by solving the characteristic equation.

If you've studied ordinary differential equations, this should be obvious. If you haven't, then you should study ODEs a little before proceeding further.
 
SteamKing said:
I'm not sure what an "armonic" oscillator is, unless you mean "harmonic".

The differential equation of the harmonic oscillator is linear and homogeneous, so the general solution is composed of the sum of all the solutions obtained by solving the characteristic equation.

If you've studied ordinary differential equations, this should be obvious. If you haven't, then you should study ODEs a little before proceeding further.
Sorry for "Armonic", In spanish it is not spelled with h, and I am so tired I did not even noticed it. Thanks a lot now I get it.
 
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