Using Complex Numbers to find the solutions (simple Q.)

[JTC]

Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function.

We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time.

Then, once done, if the forcing function was cosine, we choose the real part as the particular solution. If it was sine, we choose the imaginary solution.

Can someone say, in words, how we know this works?

I know it works because I can do it. How do we justify it? Most books just do it, but never explain why it works.

I am NOT looking for an advanced mathematical proof. I just want to know, in simple words, how we know this works.

 

[BvU]

Then, once done, if the forcing function was cosine, we choose the real part as the particular solution. If it was sine, we choose the imaginary solution.

No we don't. The observable satifies a differential equation and the observable is real. We 'choose' the real part of the solution.

A sine is a cosine with a phase difference. That way we have an amplitude and a phase as constants so that we can satisfy the boundary conditions.

Check out the damped case as well, maybe it will be a bit clearer then

 

[JTC]

No we don't. The observable satifies a differential equation and the observable is real. We 'choose' the real part of the solution.

A sine is a cosine with a phase difference. That way we have an amplitude and a phase as constants so that we can satisfy the boundary conditions.

Check out the damped case as well, maybe it will be a bit clearer then

As I understood, and worked out, if the forcing function was sine, we WOULD choose the imaginary.

And I am ONLY discussing the nature of the particular solution. I am not interested in (in the case of damping--yes, bringing that back in), the transient solution.

I am aware that I CAN work this out by sticking with the complex solution and consolidating it with a phase shift. But I am not interested in that discussion either.

I would just like to know WHY this method of using Euler's formula to extract the real or imaginary (again, NOT interested in the phase right here), works.

 

[BvU]

As I understood, and worked out, if the forcing function was sine, we WOULD choose the imaginary

Hard for me to guess how it looks -- my telepathic capabilities have proven to be quasi non-existent --, so can you post something to show that ?