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?
In: 4
Physicists, and chemists, and most scientists in one way or another, follow the fundamental idea that correlations (observable relationships of some sort) exist for a reason. They start with the simplest possible situations, where some factor X is behaving in a direct fashion with regard to some other factor Y (everything else which might matter or could matter is considered to remain constant, so only thinking of the very special case of X and Y and nothing else).
Correlation is not proof of causation, but it is pretty good evidence that something is happening that relates the two factors together in some real fashion, and the question is only one of what, not if.
There are several ways to come up with a mathematical representation of that presumed process: you can envisage the process (imagine it acts like two perfect spheres hitting each other, perhaps) and then generate the equations which would describe the paths of such objects if it were to happen, then deal with the fun of derivatives (slight changes to the conditions) and run with it; OR you could look at a series of test results and figure out what sort of relationship the two parameters are displaying (first order linear, or what?) and then play with the equations.
We mostly do not start from scratch anymore given that the main (principle) basic behaviors have already been determined and modeled/defined.
Sometimes it is just basic common sense or a basic presumption, like total energy is heat plus work. That is, energy change manifests on one way or the other (could be a third way but we don’t know about it so we start with the idea that change in energy causes change in heat and/or does some work; worry about a third concern if we find that the model fails in certain conditions).
Fundamentally, though, you look at a process, a system, and think about what might be happening, and once you have a hypothesis about what is happening, create equations which represent that hypothetical process mathematically. Then you test reality to see if it mirrors what your hypothesis-based math says it ought to.
Your question is kind of like asking how a writer comes up with a story. It comes from thinking. Some thoughts make sense or “work”, some do not. Over time, the brain gets used to finding answers that work (answers that do not work are a waste of time and frustrating, so you learn to not go that way).
No modern scientist (at least none I have ever met) is working in a knowledge vacuum. We have been shown, by the work of the old-time greats, the pioneers, how to play the game. Now, if you are asking how the first folks came up with their ideas, I cannot answer. Genius has its own way of showing itself.
Latest Answers