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.

Model.

Lets understand it though an example. You are looking into a thermodinamical system.

Ok you need to make predictions, so first you need a model. Like ideal gas: the particles are perfect spheres and they only interact through elastic collisions. There are many particles so lets introduce a way of taking averages and now we have quantities like temperature and pressure. We compress the gas and see that in our model how the particles bounce of the walls. We can calculate kinetic energy and particle density take acreages and we have equations linking volume change with temperature and pressure. We can heat up the gas at one point, and see how this model produces a heat equation.

You have a model which lets you quantify what you are interested in.

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.