I think many people would say, “it’s part of analysis!!”. After all, differential equations wouldn’t exist if functions over real numbers and their derivatives were not conceptualized. But like, woah, an introductory course in DEs wouldn’t make sense if not for our understanding of linear algebra. Functions are apparently also vectors, and derivatives are linear operators (hence they are also called differential operators).
How are functions vectors?? Let f, g, and h represent continuous functions. Well, the set of all continuous functions have an additive identity (f + 0 = f), it is closed under addition (f + g = h), they can be multiplied by a scalar (kf where k is a complex number), they have an additive inverse, etc. etc. They’re just not a finite-dimensional vector space like Euclidean space.
Derivatives are also linear operators. If D represents operation of taking the derivative of a function, then D(af + bg) = aDf + bDg for constants a and b.
Existence and uniqueness theorems regarding differential equations usually rely on concepts from real analysis (with all those wild inequalities T_T), so it also makes sense to categorize differential equations as part of analysis.
I’m so confused. We could say DEs are part of both analysis and algebra, or neither. But that’s soooo unsatisfying. I’ll just leave with the physics community’s favorite differential equations,
Figure 73 