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I'm trying to find a differential equation (first OR 2nd order) whose output could be a Fourier Series of a fundamental frequency input.... So It will be a "forced" ODE.

That is, I want

[itex] cos({\omega}t) = L[x(t)] [/itex] where

[itex]L[/itex] is some operator on [itex]x(t)[/itex] such that the solutions in [itex]x(t)[/itex] are harmonics of [itex]cos({\omega}t)[/itex] as

[itex] x(t) = a_1cos({\omega}t) + a_2cos(2{\omega}t) + a_3cos(3{\omega}t) + a_4cos(4{\omega}t) +... = \sum_{n = 1}^{\infty}a_ncos(n{\omega}t)[/itex]

...and the trick is.....I WANT IT TO SOUND "WARM".

MORE ON MY GOAL: I'm actually in my senior design course in engineering and my team is making a synthesizer for our project and we need a way to make warm sounds (e.g. sounds with timbres comfortable to the human ear.) this requires making output that sounds more interesting than simple sinusoids which are easy to generate with a simple linear difference equation (discrete, recursive definition of ODE).

So far, I've tried various things like the forced Duffing equation and the pendulum equation (which are really close to each other actually) , but nothing seems to give predictable output...

Would there be any set of operators that would allow me to change a few constants, say... [itex]a[/itex] and [itex]b[/itex] in [itex]L[/itex] to sweep across a range of neat sounds of [itex]x(t)[/itex]?

THANKS FOR READING!!!!!

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# NEED: Nonlinear ODE with harmonic output

Can you offer guidance or do you also need help?

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