What's a differential equation?

Question

What exactly is a differential equation? Pray, is $$\frac{dx}{dt}= x(x(t))$$ not a DE?

Because Arnold thinks not. In a note on page 11 of his Ordinary Differential Equations, he says, 'Differential equations are sometimes said to be equations containing unknown functions and their derivatives. This is false...'

And this is -- I remember -- exactly what we were told during our undergraduate ODE classes; I also have found this in many other places -- books, online, etc. So Arnold's comment is strange and seems like nonsense.

Who has an idea what he means, or why he's saying this?

If you take the time to browse a bit the site you will notice immediately that no one writes titles all in caps. Don't do it, please. — Mariano Suárez-Álvarez, Dec 09 '16 at 20:04
Perhaps a differential equation is just what it sounds like: an equation in terms of differentials. — Jacob Wakem, Dec 09 '16 at 20:04

score 3 · Answer 1 · answered Dec 09 '16 at 20:02

$$ \dfrac{dx}{dt} = x(x(t))$$ is not a differential equation, it is a functional differential equation. A differential equation is an equation containing an unknown function and its derivatives, all evaluated at the same independent variable(s). An equation containing $x'(t)$ and $x(\text{something else})$ is not a differential equation.

What's a differential equation?

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