For example, we have differential equations that model, for example, resonance, vibration, heat transfer, the motion of a spring, etc. How were these equations first calculated and derived, and to such a degree that we can be sure of their accuracy in modeling real life phenomena? What is the process for actually deriving them like?
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You start in simple broad observations. I’ve had to derive a bunch of these and they can start really simply. The starting point for mechanical resonance, mechanical vibrations and the motion for a spring is Newton’s second law. if I push on something, it moves. These specific ones are all related and are in the same family.
Testing and observation is vital to making sure it models real world stuff. If it doesn’t reflect the real world, then it’s back to the drawing board.
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