Is differential equation required to study real analysis?

[woundedtiger4]

Hi all,
Is differential equation a prerequisite to study real analysis (in context of baby Rudin)? And does it have any use in measure theory or Stochastic Calculus?
Thanks in advance.

 

[xaos]

i'm not sure, but i suppose you can connect these by way of distributions in functional analysis as solutions to stochastic dynamical systems especially diffusions. the idea of a distribution comes out of lebesgue integration in real analysis and depends on the idea that you can integrate up to sets of measure zero removed from the domain. this makes it a little easier to solve dynamical systems where continuity becomes an issue.

 

[HallsofIvy]

But "distributions" and "stochastic dynamical systems" are not likely to show up in a first semester class in analysis! No, I would not consider differential equations a prerequisite for real analysis.

 

[AlephZero]

Arguably it is the other way round, if the ODE course is focused on the math (existence and uniqueness of solutions, etc) rather than being a cookbook of recipes for solving particular types of ODEs.

 

[blue_raver22]

I can say that while differential equations are not a requirement for studying real analysis, they can certainly be helpful in understanding certain concepts within the subject. Real analysis is primarily concerned with the rigorous study of real numbers, limits, continuity, and differentiability, among other topics. While differential equations may not be directly related to these concepts, they do involve the use of calculus and can provide a deeper understanding of the underlying principles.

In terms of measure theory and stochastic calculus, differential equations can be useful in certain applications, such as modeling physical systems and analyzing their behavior. However, they are not essential for understanding these topics and can be studied separately.

Overall, while a basic understanding of calculus is necessary for studying real analysis, specifically differential equations are not a requirement. However, they can be a valuable tool in gaining a deeper understanding of the subject and its applications.