Okay...I think I just geeked out too much...it's a side effect from studying mathematics
Hot take:
I seem to be doing fine, for now, math is my best subject. But any day now I just feel like I'll flunk it eventually lol
Same. I actually got 96 percentile in math at my grade meaning that I scored high in every part: Geometry, Algebra, etc.

The 2 groups of math are Pure and Applied. Pure Mathematics areal, from my understanding, just the concepts. Derivatives themselves are Pure Mathematics, while Theoretical Physics are Applies Mathematics.
Can someone teach me how to solve Differential Equations, Partial Differential Equations, Matrices, and Differential Geometry

You should have to deal with casework outside of math competitions; I'm not really sure what type of mathematics you are referring to but I assume that you are attempting to solve a given mathematics problem by manually going through or using a computer program to evaluate each "case" and examine if the case provides a valid or invalid output. To be fair, I've never really heard of Casework (and less of lattice points) and I am interested in what context you're using it in. Could you further elaborate?
If you mean theory, then yes. However, there is overlap, or bridge so to speak, to applied mathematics. Which can be seen rather easily if one were to use a venn diagram for example