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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.

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    $\begingroup$ To be even more precise: a differential equation is a submanifold of a certain finite degree jet bundle, with a solution being a section. But do you learn anything from it? Pedagogically it is often preferable to "lie" a little to get the intuition across, rather then losing the student from the get-go. $\endgroup$ Apr 15, 2011 at 17:52
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    $\begingroup$ @Willie Wong: Arnold's comment is rather antagonistic, but I don't feel he is complaining about lack of precision in the literature so much as he is trying to give a different intuition. The idea that Arnold goes for is that a differential equation is a vector field. It's intuitively clear that a vector field has a flow, so he defers the proofs of existence/uniqueness. Certain parts of ODE theory become much more accessible this way (e.g. Lyapunov functions become much clearer). $\endgroup$
    – Sam Lisi
    Apr 17, 2011 at 13:30
  • $\begingroup$ @Sam: I completely agree. The point I was trying to make is that said intuition may not be sufficiently intuitive to students without the proper preparation. When students only think of functions on $\mathbb{R}^n$ as an assignment of a value to $n$-tuples, the introduction of geometry may just be a tiny bit more work than other authors are willing to go through. $\endgroup$ Apr 17, 2011 at 14:25

3 Answers 3

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"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.

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  • $\begingroup$ By y'(t) = y(t - 1) do you mean y'(t) = yt - y or is y a function of (t - 1)? If y is a function of (t - 1) then I think it's not a diff eq because everything isn't just a function of t. If you mean yt - y then I don't know why it isn't a differential equation, looks like one to me - of the form y'(t) = F(t,y). In any case can you recommend any other books that approach this subject in a similar fashion to Arnold, I just like alternative viewpoints of the same material. It may be my naivety but none of the other books I quotes approach this subject in the way Arnol'd does. Thanks a lot. $\endgroup$
    – sponsoredwalk
    Apr 15, 2011 at 22:04
  • $\begingroup$ @sponsoredwalk: yes, certainly y as a function of t-1 is meant (i.e. the composition of y with the function which sends t to t-1), not the pointwise product. $\endgroup$ Apr 16, 2011 at 0:11
  • $\begingroup$ So why would one want to rule out x(x(t)) and y(t-1)? $\endgroup$
    – Simon
    Apr 16, 2011 at 9:53
  • $\begingroup$ Because in x(x(t)) what you're looking at is basically x(3), or x(16) etc... x is a function of some value (like 3, or 16) & not a function of an element of some domain (i.e. t). to be crystal clear, x(x(t)) = x(3) means the inside x(t) is 3, so x(t) = 3 → x(x(t)) = x(3). In y(t - 1) you see that y is a function of something other than t & it fails to meet the requirements of the definitions given. That's what I think anyway, could be wrong. $\endgroup$
    – sponsoredwalk
    Apr 16, 2011 at 11:52
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    $\begingroup$ @user10478: I don't think there is a one-size-fits-all answer to your question. It depends on the person and on what exactly you mean by linear algebra and multivariate calculus. I still agree with what I said before: Arnold's book is a great book, but I think there is great benefit to having it as a complement to a more traditional ODE book. A book that is more accessible but still quite demanding and filled with beautiful insight is Strogatz's Nonlinear dynamics and chaos. $\endgroup$
    – Sam Lisi
    Apr 29, 2019 at 1:44
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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.

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    $\begingroup$ Awww, you are not being annoyingly precise. :-) $\endgroup$ Apr 15, 2011 at 17:48
  • $\begingroup$ Let me be annoying: what kind of function should $f$ exactly be allowed to be? $\endgroup$ Apr 15, 2011 at 19:47
  • $\begingroup$ @wildildildlife: whatever kind is necessary for your application. It doesn't seem productive to me to place arbitrary constraints on $f$ ahead of time. $\endgroup$ Apr 15, 2011 at 20:50
  • $\begingroup$ I don't know why this rules out Arnold's example though, can't you just form dx/dt - x(x(t)) = 0 i.e. f(t,x,x') = 0 ? $\endgroup$
    – sponsoredwalk
    Apr 15, 2011 at 22:02
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    $\begingroup$ @sponsoredwalk: what is $f$ in this situation? (The input into $f$ is not the entire function $x$ but only the number $x(t)$.) $\endgroup$ Apr 15, 2011 at 22:22
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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:

  1. Each $I\subseteq \text{proj}_1 U$ (i.e. $I$ is a subset of the projection of $U$ onto the first coordinate direction);

  2. $\text{proj}_N u\neq 0$ for some $u\in U$ (so that the ODE is actually $N^{th}$ order); and,

  3. $(x,y(x),y^\prime (x), \ldots , y^{(N)}(x))\in U$ 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?

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  • $\begingroup$ I see. You don't get the sense of my answer. Long story short, all the previously given definitions may be stated in a better form. $\endgroup$
    – Pacciu
    Jun 14, 2017 at 14:12
  • $\begingroup$ It seems this post attracted somehow a lot of attenction in those days, even if I cannot tell why. I gratefully thank users who have edited this post, but please be respectful: try not change what I wrote in what you wanted me to write. Thank you. ;-) $\endgroup$
    – Pacciu
    May 21, 2019 at 1:04

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