In a differential equation, is recursion allowed?

Question

Take the following ordinary differential equations.

$$y'(x^2) + y(x) = x$$

$$y''(y'(y(x))) = x^{x^x}$$

$$y''(5) + y(x) = x$$

Is there anything wrong with any of these as differential equations? I know that nonlinear equations can result from terms consisting of various derivatives of $y$ multiplied together. For instance: $y(x)y'(x) = x$. However, these seem almost recursive. I do not know whether or not they are valid.

Answer 1

In your original post, they were not valid for a somewhat technical reason. Either you treat $y$ as a function, in which case $y'$ and $y''$ are both functions as well, or you treat $y$ as a variable and not a function. However, it was relatively obvious what you actually meant, and indeed it is very clear in your current version of the question. I'll leave my original explanation of that specific issue in the next section.

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In the former case, "$y$" does not represent a real/complex number and so your differential equations should have been written:

$y'(x^2)+y\color{red}{(x)} = x$.

$y''(y'(y\color{red}{(x)})) = x^{x^x}$.

$y''(5) + y\color{red}{(x)} = x$.

In the latter case, where $x,y$ are both variables varying with respect to some parameter (which is a common interpretation in classical mechanics), your equations would mean a totally different thing:

$y' \times x^2 + y = x$.

$y'' \times y' \times y = x^{x^x}$.

$5 \times y'' + y = x$.

It is clearly crucial to know which interpretation is used, and also crucial to be consistent. It is certain that your example of "a nonlinear equation resulting from terms consisting of various derivatives of y multiplied together" uses the second interpretation:

$y \times y' = x$, where $x,y$ are variables.

Which if written using the former notation would be:

$y(x) \times y'(x) = x$, where $y$ is a function.

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As I implied in my original post, you are justified in calling your equations differential equations.

It is true that the most commonly known differential equations generally relate variables and their instantaneous derivatives, and so there is no need of an explicit symbol for the parameter (usually time). This is in fact the reason for the second notation shown above.

However, I can come up with scenarios where there is a natural necessity for differential equations that relate variables to derivatives at other points in time. For example, homeostatic mechanisms have feedback loops that maintain some desirable state, such as core body temperature. But these mechanism kick in after some lag, so the differential equations to model them will have to take that into account to be accurate.

Answer 2

They are "valid". The first two are compositions of functions. The last one is a little weird because $y''(5)$ is a constant $\Rightarrow$ $x-y$ is a constant $\Rightarrow$ is the equation of a line $\Rightarrow$ $y''=0$. The second one probably has no hope of being solved.