How do scientists know which differential equation suits the most for describing the system?
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Most of the time, they apply a general law (Newton’s second law, Schrödinger’s equation, general relativity, Maxwell’s equations…) to a specific problem, often have to make some assumptions and some calculations in order to simplify it, and they get a differential equation.
As to how these global laws where found, it does take a genius for that. Some were found by making the correct assumptions, some appear to have been pulled out of thin air.
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