# differential equations and physical intuition

Often 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?

$$\vec{\nabla} \centerdot \vec{E}(\vec{r}) = 4 \pi \rho(\vec r).$$
Solving this differential equation for the electric field $\vec{E}$ 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 $\vec{r}$. Then you can choose your surface of integration to make the dot product trivial (a sphere centered at $\vec{r} = 0$). Then your partial differential equation becomes a relatively easy regular differential equation in one variable.