Definition of a Differential Equation?
Here is one definition of a differential equation:
"An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a differential equation (DE)" (Zill - A First Course in Differential Equations)
Here is another:
"A differential equation is a relationship between a function of time & it's derivatives" (Braun - Differential equations and their applications)
Here is another:
"Equations in which the unknown function or the vector function appears under the sign of the derivative or the differential are called differential equations" (L. Elsgolts - Differential Equations & the Calculus of Variations)
Here is another:
"Let $f(x)$ define a function of $x$ on an interval $I: a < x < b$. By an ordinary differential equation we mean an equation involving $x$, the function $f(x)$ and one of more of it's derivatives" (Tenenbaum/Pollard - Ordinary Differential Equations)
Here is another:
"A differential equation is an equation that relates in a nontrivial way an unknown function & one or more of the derivatives or differentials of an unknown function with respect to one or more independent variables." (Ross - Differential Equations)
Here is another:
"A differential equation is an equation relating some function $f$ to one or more of it's derivatives." (Krantz - Differential equations demystified)
Now, you can see that while there is just some tiny variation between them, calling $f(x)$ the function instead of $f$ or calling it a function instead of an equation but generally they all hint at the same thing.
However:
"Let $U$ be an open domain of n-dimensional euclidean space, & let $v$ be a vector field in $U$. Then by the differential equation determined by the vector field $v$ is meant the equation $x' = v(x), x \in U$.
Differential equations are sometimes said to be equations containing unknown functions and their derivatives. This is false. For example, the equations $\frac{dx}{dt} = x(x(t))$ is not a differential equation." (Arnold - Ordinary Differential Equations)
This is quite different & the last comment basically says that all of the above definitions, in all of the standard textbooks, are in fact incorrect.
Would anyone care to expand upon this point if it is of interest as some of you might know about Arnold's book & perhaps be able to give some clearer examples than $\frac{dx}{dt} = x(x(t))$, I honestly can't even see how to make sense of $\frac{dx}{dt} = x(x(t))$. The more explicit (and with more detail) the better!
A second question I would really appreciate an answer to would be - is there any other book that takes the view of differential equations that Arnold does? I can't find any elementary book that starts by defining differential equations in the way Arnol'd does & then goes on to work in phase spaces etc... . Multiple references welcomed.
"When I use a word," Humpty Dumpty said, in a rather a scornful tone, "it means just what I choose it to mean—neither more nor less."
I think Arnol'd is correct, but I think he is being unnecessarily confrontational about it. All the books on your list that I am familiar with nearly immediately jump to a more precise formulation that a differential equation is one of the two following things: \[ y^{(n)}(t) = F(t, y(t), y'(t), \dots, y^{(n-1)}(t) ), \] or \[ G(t, y(t), \dots, y^{(n)}(t)) = 0. \]
Here is another example of an equation that I would not want to call a differential equation: \[ y'(t) = y(t-1). \] This meets the heuristic definition, but fails to be of the form I specified above (or of the form Arnol'd considers).
I now see that Qiaochu has written nearly the same thing above.
btw, I think Arnold's book is fantastic, but should be complemented with a more standard treatment of ODE, if only so that you know what everyone else knows in addition to the topics Arnold focuses on.
EDIT: To answer the 2nd half of the question, I don't know of any books that are as geometric as Arnold. IMO, the big strength of his book is that he makes the geometric intuition jump out at the reader, and downplays the analytical side of things. This complements the more traditional books that focus on the analytical aspects (and on explicit solutions) and lose all the geometry.
Arnold has another book that is somewhat more advanced, Mathematical Methods of Classical Mechanics. I think it's another great book, though it's hard to read. He also has a book called Geometrical methods in the theory of ODE. This is also a more advanced book, so it is not one you want to look at yet.
A book that I found very compelling was Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra. It's more analytical than Arnold, but is more geometric than most.
EDIT 8 years later: Let me add a recommendation for Strogatz's Nonlinear dynamics and chaos. I think it's a beautiful book and wish I could go back in time and give it to my younger self.
Arnold simply means that most books are not being precise. A slightly more precise version of the first few definitions is that a differential equation (in one variable) is an equation of the form $f(t, x, x', x'', ...) = 0$. This rules out Arnold's example.
When I was a student I was taught the following definition:
Let $N\in \mathbb{N}$, $U\subseteq \mathbb{R}^{N+2}$ and $F:U\to \mathbb{R}$.
Then the $N^{th}$ order ordinary differential equation (in implicit form) corresponding to $F$ is the problem of finding all the non-degenerate intervals $I\subseteq \mathbb{R}$ and all the functions $y:I\to \mathbb{R}$ such that the following hold:
Each $I\subseteq \text{proj}_1 U$ (i.e. $I$ is a subset of the projection of $U$ onto the first coordinate direction);
$\text{proj}_N u\neq 0$ for some $u\in U$ (so that the ODE is actually $N^{th}$ order); and,
$(x,y(x),y^\prime (x), \ldots , y^{(N)}(x))\in U$ and $F(x,y(x),y^\prime (x),\ldots ,y^{(N)}(x))=0$ for each $x\in \text{int }I$.
This problem can be denoted for short as: $$F(x,y,y^\prime, \ldots ,y^{(N)})=0\; .$$
If the function $F$ is of the type: $$F(x,y_0,y_1,\ldots ,y_N)=f(x,y_0,y_1,\ldots ,y_{N-1})-y_N$$ then the differential equation is said to be in normal (or explicit) form and it can be denoted for short as: $$y^{(N)}=f(x,y,y^\prime ,\ldots,y^{(N-1)})\; .$$
What do you think about it?