Definition of a Differential Equation?

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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
ƒ 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 ƒ(x) the function instead of ƒ or calling it a function instead of an
equation but generally they all hint at the same thing.

However:

"Let U be an open domian 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 e U.

Differential equations are sometimes said to be equations
containing unknown functions and their derivatives. This is
false. For example, the equations 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 you care to expand upon this point if it interests you as some of you
might know about Arnold's book & perhaps give some clearer examples than
dx/dt = x(x(t)), I honestly can't even see how to make sense of dx/dt = x(x(t)).

A second question I really would 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... . Merscieee :cool:

Fremen

It seems as if Arnold is suggesting that x(x(t)) doesn't define a sensible vector field, but I'm not sure why that would be. Maybe someone on stackexchange will know. If you find out, I'd be interested to hear.

The other textbook definitions could be fixed rather easily - by changing "relation" to "admissable relation", then constraining "admissable relation" to agree with Arnold's treatment. The other books are technically "wrong", but really they're just giving simplified explanations for pedagogical purposes. I would say this is a very sensible thing to do. There's no good reason to hit a new student with all the technical machinery in a subject. This is why Riemann integration is taught before Lebesgue integration, and why random variables are introduced as "the outcome of an experiment", instead of being introduced as measurable functions.

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"Lang"?

I came across this last night & meant to post but was too tired:

---

Let Ω ⊆ ℝ ⁿ ⁺ ¹ be a domain and I ⊆ ℝ be an interval.
Let F : I × Ω → ℝ be a function defined by
(x, z, z₁, . . . z_n ) ↦ F (x, z, z₁ , . . . z_n )
such that F is not a constant function in the variable z_n .

With this notation and hypothesis on F we define the basic object in our
study, namely, an Ordinary differential equation.

Definition 1.1 (ODE) Assume the above hypothesis. An ordinary
differential equation of order n is defined by the relation

F(x, y, y⁽¹⁾ , y⁽²⁾ , . . . y⁽ⁿ⁾ ) = 0, (1.1)

where y⁽ⁿ⁾ stands for nth derivative of unknown function x ↦ y(x)
with respect to the independent variable x.

...


3. (Arnold) If we define an ODE as a relation between an unknown
function and its derivates, then the following equation will also be an
ODE.

dy
--- = y ◦ y(x) (1.2)
dx

However, note that our Defintion (1.1) does not admit (1.2) as an ODE.
Also, we do not like to admit (1.2) as an ODE since it is a non-local
relation due to the presence of non-local operator ‘composition”. On the
other hand recall that derivative is a local operator in the sense that
derivative of a function at a point, depends only on the values of the
function in a neighbourhood of the point.

link

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What do you think?

Fremen

This reminds me of the setup for the implicit function theorem. Not sure where that gets you though.

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Stackexchange post on the topic. I was reading x(x(t)) off all wrong.

If anyone can recommend a book that approaches the subject in a
similar manner to Arnol'd please share! :cool: