Differential equation is one of the many research directions of basic mathematics and applied mathematics, with different emphasis. Basic mathematics mainly focuses on the uniqueness of the solution and whether it is a positive solution, the structure of the solution, applied mathematics mainly focuses on the trajectory of the solution, dynamic system and so on. Computational mathematics is also involved, mainly focusing on the numerical solution of the equation. For example, finite element method is a new method to solve equations, which is not new, but it is a new research method in modern times. We should know that most differential equations can't be solved by classical differential equations, so we can only study them by other methods, and indirectly get the properties of solutions or relatively accurate approximate solutions.

Pure mathematics is not just number theory. Number theory is an aspect of pure mathematics, which includes cryptography, information security and so on. There are also algebraic research directions of algebra, such as algebraic semigroups, lie groups and equations, such as the differential equations you mentioned, which is also a research direction of pure mathematics, and there are many other more detailed research directions.

With the rapid development of modern mathematics, the differentiation of majors above master's degree will become more and more detailed, not just a few majors and research directions. I hope you can find a related major that interests you.

I hope my answer is helpful to you.