differential equations and physical intuitionOften when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My question is: If you are a pure mathematician studying differential equations, do you have to be good at physics (biology, finance) too?

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Rihanna Blanchard · Accepted Answer

Step 1By looking at physical phenomena, you&#039;ll see an application of a (partial) differential equation expressed in a particular coordinate system, or with particular boundary conditions.The solutions for these applications will likely be a subset of a more general solution.Take Gauss&#039;s Law:            ∇      →        ⋅            E      →        (            r      →        )  =  4  π  ρ  (            r      →        )  .Solving this differential equation for the electric field             E      →        might be made much easier if you consider that the electric field is derived from a scalar potential function that depends only on the magnitude of             r      →       . Then you can choose your surface of integration to make the dot product trivial (a sphere centered at             r      →        =  0). Then your partial differential equation becomes a relatively easy regular differential equation in one variable.Step 2But this might be a bit divorced from what you need or want. Physicists use differential equations to explain something: electromagnetic fields, mechanics, heat flow, etc. Once they have the model (the equation) they apply it, and see if it matches reality. If you&#039;re staying within the realm of mathematics, then you might not want to do that, but instead rely on formal proof, or other &quot;things you know&quot; from related areas of mathematics. Using physical intuition in pure mathematics might lead you to a wrong conclusion, because your application might not be correct.

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