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I actually asked a version of this question before, here: What is the formal, rigorous definition of a differential equation?. However, I should have asked what an equation is, first, which I did here: What is the formal definition of an equation?. But now that I have asked the question of what an equation is, and gotten a very detailed and rigorous answer, I want to revisit my original question. I want to know what is the formal and rigorous definition of a differential equation. I don't think it is simply a functional equation, like $f^2=f+1$. That is an equation involving real functions, but it is not a differential equation. So, then, what is the definition of differential equations?

user107952
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    What makes the definition of ODE we learn in class non rigorous? – Kurt G. Feb 17 '24 at 22:43
  • @KurtG. A differential equation is not an equation relating functions and their derivatives. For example, $\dot x=x(x(t))$ is not a differential equation. – John Douma Feb 17 '24 at 23:22
  • VI Arnold defines an ordinary differential equation as determined by a vector field. See "Ordinary Differential Equations", chapter 2 by VI Arnold. – John Douma Feb 17 '24 at 23:24
  • There is a good answer given for ODEs given by Pacciu here. A similar answer works for PDEs you just have to use partial derivatives and ibstead of an open interval use an open connected subset in the variable space. – Moishe Kohan Feb 18 '24 at 01:23
  • @JohnDouma I learned the Arnold definition. It is rigorous. – Kurt G. Feb 18 '24 at 04:57
  • @KurtG. I upvoted this question because an internet search for the definition of an ordinary differential equation does not yield a rigorous definition. The definition given by Arnold is surprisingly elusive. – John Douma Feb 18 '24 at 08:53
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    @user107952: Does this answer it https://math.stackexchange.com/questions/33153/definition-of-a-differential-equation?rq=1? What is missing there that is different? https://math.stackexchange.com/questions/33153/definition-of-a-differential-equation?rq=1 – Moo Feb 18 '24 at 11:53
  • @Moo: If you limit yourself to ODEs, then this is indeed a duplicate, but the link does not provide a definition of a PDE. See my comment above. In tags, OP says "ODEs", but not in the body of the question. – Moishe Kohan Feb 18 '24 at 14:26
  • @MoisheKohan: I can only go by what the question contains. I did not flag it, but it seems like a dupe to me. Regards. – Moo Feb 18 '24 at 16:26

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