Harmonic oscillator derivation

[pabilbado]

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:

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.

 

[SteamKing]

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:

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.

 

[pabilbado]

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.