Q:

General form of a differential equation.

In general, a differential equation can be represented as
$$f(x,y, y', ..., y^{(n)}) = 0$$
is it also then valid to say that a differential equation can be represented as
$$f(x, y) = 0,$$
since $y^{(i)}$ depends entirely on $y$?

A:

In the first representation, we have a function $$
f:\Bbb R^{n+2}\to\Bbb R
$$
which is not the case for the second case.
Such functions are far more easy to manipulate than general functionals such as the second equation suggests.

