How do scientists know which differential equation suits the most for describing the system?
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A combination of theory and trial and error. For a given system, we’ll usually have a theory of what’s going on, and can apply equations from similar problems that have been solved in the past. We will fit that equation to the data and see how closely it matches. If it’s not a good match, it either means that our theory is wrong, or (more commonly) that there’s something else happening at the same time on top of what we thought was happening. Sometimes, we can work backwards by finding an equation that fits the data, and then hypothesizing what kind of behavior is described by that data.
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