If I wanted to write the most general possible form for an ordinary differential equation of some integer order $n$, what would that look like? I wish to compare two different types of differential algebras but I cannot meaningfully determine what all differential equations of some order might look like at an abstract level. I can write all linear equations of an arbitrary order (it's a pretty trivial form to write), but I am not sure how to do all nonlinear equations. I know from a previous question I asked that $y$ is the dependent variable and $x$ is the independent variable that things like $y(y(x))$ are allowed. With that in mind, something like $f(x,y,y',y'',y''') = 0$ (where f is any function) would not be a valid form for all 3rd order ordinary differential equations.

So then using my potential form above as a model to improve upon: what do I need to change to incorporate the recursion?

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    $\begingroup$ Hmm you first need to define "order". Normal differential equations can be thought of as an equation relating $f,D(f),D^2(f),...$ uniformly at every point (namely the same equation for every input), where $D$ is the differential operator. Then it makes sense to define order as the highest exponent of $D$. It no longer makes much sense in your case. $\endgroup$ – user21820 Apr 2 '17 at 6:25
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    $\begingroup$ You're making a false assumption. Order is not well-defined simply because you haven't defined it. Mathematicians have defined order for differential equations of exactly the form I've described. So if you want to talk about order for differential equations not of that form, you had better define what you mean. Otherwise your question is not precise. $\endgroup$ – user21820 Apr 2 '17 at 6:34
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    $\begingroup$ To clarify what I mean, Wolfram Mathworld and other places define a differential equation to be "an equation that involves the derivatives of a function as well as the function itself", which includes your kind. Of these some are called ODEs, PDEs, DDEs and so on. Order has been defined for these, such as here for ODEs. Since nobody has defined order for your kind, you'd have to define it. The reason people don't consider your kind of differential equations is because they just aren't needed in practice. $\endgroup$ – user21820 Apr 2 '17 at 7:42

If I wanted to write the most general possible form for an ordinary differential equation of some integer order n, what would that look like? $ \def\rr{\mathbb{R}} $

An ordinary differential equation of order $n$ is defined as an equation expressible in the form "$f(x,y,{D_x}(y),{D_x}^2(y),\cdots,{D_x}^n(y)) = 0$" where $x,y$ are real variables (with $y$ varying with respect to $x$) and $f$ is a function from $\rr^{n+2}$ to $\rr$. It includes equations like "$x+y y' = y''$", but clearly does not include things like "$g'(x^2) = g(x)$" where $g$ is a function from $\rr$ to $\rr$.

With that in mind, something like $f(x,y,y',y'',y''') = 0$ (where f is any function) would not be a valid form for all 3rd order ordinary differential equations.

As stated above, it is valid (for ODEs) when interpreting your $y',y'',...$ as $D_x(y),{D_x}^2(y),...$. But if you want to define order for non-ODEs, you're going to have to do it yourself because it seems no one has bothered about them (no known practical use).

Order is still well-defined.I don't see how allowing y'''(y(x)) = 0 as a differential equations makes it any harder to tell that both it and y(y'''(x)) = 0 are third order differential equations.Order simply indicates the highest order derivative present in the equation.

That only works if you define what you mean by "equation", which is not as trivial as you may think. You could define it exactly like in defining syntax of higher-order logic:

  • "$x$" is an object-term.

  • "$y$", "$y'$", "$y''$", ... are $1$-input function-terms.

  • There is an object-term for each real number.

  • There is a $k$-input function-term for each function from $\rr^k$ to $\rr$.

  • "$f(t)$" is an object-term for every function-term $f$ and object-term $t$.

  • "$t = 0$" is a differential equation for every object-term $t$.

And of course, there are no other object-terms or function-terms other than those generated by the above rules. Now it should be clear how you can define order to express what you want; define the order of a differential equation to be the highest order of derivative of $y$ that is used (symbolically) as a function-term in it.

But I doubt the usefulness of such a definition. "$y(y'(y(x^2))) = \sin(x)$" would be a first-order DE under such a definition, but order seems to have no useful consequence, unlike in the case of ODEs. In the end, definitions are only useful if you can prove useful theorems about the defined objects.

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    $\begingroup$ @TheGreatDuck: Indeed, differential equations are just special kinds of functional equations, because functional equations are literally arbitrary equations involving functions where the goal is to find what functions satisfy them. Occasionally, people may be interested in whether there are differentiable functions that satisfy a certain functional equation, such as $f(f(z)) = \exp(z)$. $\endgroup$ – user21820 Apr 4 '17 at 15:23
  • $\begingroup$ @TheGreatDuck: That's interesting, since it at least shows that people have thought about such weird differential equations before even if they are not useful. And you're welcome! $\endgroup$ – user21820 Apr 4 '17 at 15:38
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    $\begingroup$ @TheGreatDuck: If you still have that textbook, it might be interesting to really note down their example differential equation in your other question (even if just as a comment) as evidence that it's really been considered before for some application and not just a purely abstract notion. $\endgroup$ – user21820 Apr 4 '17 at 15:49

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