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I am trying to understand the boundaries of the definition of an ODE, for simplicity, say, of first order. I understand it should be of the form: $F(x,y,y')=0$. But this includes equations of the form:

  1. $y(x)=f(x)$
  2. $y'(x)=f(x)$

My guess is that the equation 1 cannot be considered a differential equation (since it is not "differential") and that equation 2 can be considered a differential equation. So for example: $y'=0$ is an ODE.

First question: Am I right?

Second question: Why is it not specified in the definition? (something like: "$F(x,y,y')$ must contain at least one occurrence of $y'$")

user135172
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  • From the General Definition section of the page you cite, why would you reject the existence of ordinary differential equations of order zero? – Eric Towers Feb 09 '25 at 09:11
  • That definition is the one that allows to apply seamlessly the existence and uniqueness theorem by Picard & Lindelöf. Question for you: is $y'(x)=y(y(x))$ an ODE? Spoiler. – Kurt G. Feb 09 '25 at 09:34

1 Answers1

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My guess is that the equation 1 cannot be considered a differential equation (since it is not "differential")

I would argue it could be - but only in a degenerate way. I would also in that same argument say something in the vein of "why would you care" or "why would you use techniques for ODEs on something not even invoking a derivative".

Why is it not specified in the definition? (something like: "$F(x,y,y')$ must contain at least one occurrence of $y'$")

For probably a couple of reasons:

  • When invoking it in the definition of $F$ as such, we're implying on some level that $F$'s that actually use $y'$ are probably the central object of study - if just by the context of the study.

  • Because we very very rarely actually study the structure of the function $F$ itself: defining its particulars and limitations, for an object so general, really isn't helpful for anyone. We're really just saying "we're studying relations that involve an input $x$, a function of that input $y$, and derivatives of said function, set equal to zero" in mathematical language. To delve even the tiniest bit deeper into studying ODEs, we tend to have to immediately impose further restraints or look at more specific forms. ... That is to say, it really is just a definition a professor would give on day $1$ to say "we're studying differential equations" and little else.

PrincessEev
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