How do scientists know which differential equation suits the most for describing the system?
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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.
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