Talk:Differential equation

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Old talk moved (along with page) to Talk:Ordinary differential equation. -- Walt Pohl 20:54, 23 Jan 2005 (UTC)

Contents

[edit] Bias

As an objective student seeking a broad theoretical introduction to differential equations, I get a pretty negative vibe from the section "Directions of Study". It seems that a "pure" mathematician wrote this and has a fairly negative view of applied mathematics - I don't think an applied mathematician would really not care "whether these approximations really are close to the actual solutions." I had a super purist theoretical math professor a few courses ago that sounded like this, and he essentially demonized applicational math. my 2 cents. —Preceding unsigned comment added by 130.91.131.112 (talk) 15:52, 26 November 2007 (UTC)

As a physicist, I agree. Strictly speaking, the text is correct, but the equivalent statement that pure mathematicians are only concerned with the validity of solutions and not with anything that has an application is equally unflattering. I changed it to a classification of fields rather than people to avoid bias. —Preceding unsigned comment added by 138.67.37.68 (talk) 19:24, 26 November 2007 (UTC)

[edit] Text copied for keywords

Hello. I understand that other sites can use information from this site, and my site is a free information sharing wiki also. I've copied the main text of this front page and put it in a comment in my web site template (www.exampleproblems.com link), because it has lots of keywords and math phrases. I hope that is ok. -thanks -Tbsmith

[edit] Removing reference

I removed a reference that seems to have been added for no other reason than to allow a link to the bookseller's page. The reference was added by someone who doesn't seem to have contributed to the article in any way; and the same reference was added to Ordinary differential equation by the same person (who doesn't seem to have contributed to that article, either). The reference doesn't seem to be well known or widely available, and I find it quite doubtful that it was used at all in writing or editing this article, or used meaningfully in fact-checking. In anyone objects to this removal, please discuss here. - Ruakh 22:53, 13 August 2005 (UTC)

[edit] Vote for new external link

Here is my site with differential equations example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php?title=Ordinary_Differential_Equations

[edit] Intended audience

"In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables"

Who is this article aimed at? Seems to be another mathematics article that is written by a mathematician for another mathematician of the same level of knowledge. If differential equations require a certain level of knowledge of mathematics to be understood then this would be nice to be indicated on the page.

[edit] Linearity

"A differential equation is linear if it involves the unknown function and its derivatives only to the first power; otherwise the differential equation is nonlinear."

I am not sure this is the correct definition. What about eqn's such as |dy/dx| + |y| = x? Isn't this non linear. (Here |.| is the modulus function.) What about y(dy/dx) = x. I though linear equations were defined as those which are of the form: Ay^(n) + By^(n-1) + ... Cy + D = 0 where A,B,...,C,D are functions of x. Not sure how this classifies a PDE. Can someone please clarify?

Your definition is indeed the correct one, and works fine for ODEs. PDEs linearity is analogous, though for a better understanding of linearity one should consider that a ODE or a PDE can both be written as Pf=g where f and g are suitable functions on R^n and P is a differential operator. Evaluating the linearity of both is actually testing the linearity of the differential operator.

(Antonius Block 23:39, 27 November 2006 (UTC))

[edit] About the Image

The caption reads

An illustration of a differential equation. The arrows show how the differential equation locally influences a state, while the lines display how specific solutions are determined by starting conditions (red dots).

I agree to the point of red dots and their relation to initial conditions. But what is meant by the arrows show how the differential equation locally influences a state??? To my knowledge these lines are the so-called integral curves and the arrows define the direction field of that particular differential equation which is used to solve that equation approximately by plotting several integral curves. This method is called isocline or isoclyne or whaterver (sorry I don't know maths terminology in English very well).

Am I right? If that is the case, could someone please correct the caption? RokasT 19:16, 9 January 2007 (UTC)

The image is gone as it was copyvio. Can someone make a new image? User A1 05:55, 2 February 2007 (UTC)
It's not clear to me why the image was deleted. I asked the person responsible for clarification at commons:User talk:Bryan. -- Jitse Niesen (talk) 06:59, 2 February 2007 (UTC)


I asked for it to be deleted (on IRC, didn't know how to delete stuff from commons) because its the direct rip of the samples provided with apples grapher application, no actual work has been done other than "print screen". see User talk:Romansanders User A1 07:35, 2 February 2007 (UTC)

[edit] Made a new graph

Hello,

I made a new image for this article

as I was responsible for the deletion of the other image (due to copyright issues) i felt i should replace it with an image with a clearer licence. Can someone add this to the article? Thanks User A1 05:48, 24 February 2007 (UTC)


Sure, I'll add it in... thanks a lot!!, and feel free to contribute more.. this article doesn't seem to do justice to the topic. Danski14 16:21, 24 February 2007 (UTC)

[edit] Rise in importance during 20th century

This section contains interesting material, but in my opinion, it does not belong to this article. First, it deals exclusively with the war effort (World War II, that is) in one country, USA, not with the general development of differential equations. So the title is rather misleading. Moreover, it describes some numerical approaches to differential equations, but nothing else. Differential equations had been important at least since the time of Newton, and indeed in earlier times people like Euler worked on problems of ballistics, among other things. Can anyone think of a suitable article to which this section can be moved? Something having to do with applied mathematics, war effort, or numerical methods, perhaps? Arcfrk 02:51, 15 May 2007 (UTC)

Actually, there is a point here; solutions to differential equations became evaluable whether or not they could be solved in closed form, and that change propagated world-wide shortly after the war. A {{main}} to History_of_computing_hardware#Pre-1940_analog_computers or somewhere like that; cutting it down to a summary; and, yes, changing the title would make this quite bearable. Septentrionalis PMAnderson 05:08, 17 May 2007 (UTC)
This section seems to be about the rise of certain mathematical tools into areas of engineering like automatic control, and related advances in computing. It's odd, to say the least, to put it into this article. The same points can be made for other parts of mathematics not used so often in engineering in early 20th century. One problem with this section is that it wasn't written by a historian, but rather by someone, with his/her fascination with certain aspects of the history. So one ends up with a mainly mathematical article with a historical section with certain rather obvious biases and inaccuracies. Not a satisfactory state of matters, in my opinion. --C S (Talk) 09:43, 17 May 2007 (UTC)
It appears that the section has now been deleted, and since it is not reproduced on this talk page, its hard to tell what the fuss was about. linas 15:44, 20 May 2007 (UTC)
No, it was simply renamed 20th century applications, and is still very much out of place. Arcfrk 02:13, 21 May 2007 (UTC)
There seems to be some argument over the name of the cannon discussed in the article. It is either Big Bertha or the Paris Gun. I believe that both guns had the capacity to achieve high enough altitudes. Possibly both should be mentioned? Bygeorge2512 17:58, 21 May 2007 (UTC)
The article quotes it as however, such as Germany's giant cannon with singular usage. The Paris gun reached up to 100km distance, with the Big Berthas reached only 12km distance. The Berthas were used mainly to reduce the forts around Liege and Namur during 1914, the Paris Gun, and other railroad guns were mainly used to shell Paris from a distance. I think that the cannons should be made plural, and include both the Berthas and the Paris Gun. Spirits in the Material 18:04, 21 May 2007 (UTC)

I moved this section to History of numerical solution of differential equations using computers. Arcfrk 08:57, 2 June 2007 (UTC)

Well, as the author of the text in question here, I think it was a good idea of Arcfrk to move the historical section out to another area rather than delete it altogether. I did feel that for the sake of focus the main entry should stick to the basic definition and the mathmatical functions. It is a good thing to stick to the principle of "bare bones" for the main entries. However, I did not see another place to make the important connection between the development of computing machines and the application of differential equations. This relationship is not often discussed or understood but it is not theoretical since the history is well documented. Reading Wiener's "Cybernetics" and anything about von Neumann's contribution to the development of computing machines for dynamics of ballistics and anti-aircraft weapons during World War II will show this to be true. As for there being "certain rather obvious biases and inaccuracies," however, please point them out specifically. If what is meant by bias is that development in other nations at other times has not been included, then it should be included. The crux of the entry is that the need for fast computing of differential equations for stochastic behaviors vastly accelerated the development of the computing machine you're reading this on. We could go as far back as cybernetic mechanisms on steam ships in the 1800's to prevent oversteering, or as far forward as landing a rover on Mars if we were to round out the story. My background is in telecommuncations but teaching it I must also teach the history. Thanks to anyone who improved the text. --andytalk 22:09, 2 June 2007 (UTC)

[edit] Is differential for the geniuses only?

Ladies and gents, let's be reasonable here. Wiki was made to simplify meanings and definitions. I have nothing against the "authors" of this article but is there any way to "simplify" the terms and explanations for this? 'Coz it's like im reading an online book. Thanks! One Name124.104.141.201 06:28, 2 June 2007 (UTC)

Well, there is little doubt that this article is still in its infancy and requires a lot of work, which is reflected by its 'Start' class rating. However, you seem to be mistaken with regards to the purpose of Wikipedia. As pointed out in many places, for example, at Wikipedia:Five pillars, Wikipedia is an encyclopaedia, and as such, its purpose is to sum up the human knowledge, rather than 'simplify meanings and definitions' (see, however, the Simple English edition for attempts in these directions). And encyclopaedias are real books, not CliffsNotes versions. Incidentally, the following phrase is attributed to Albert Einstein:
A physical theory should be made as simple as possible, but not any simpler.
(Variants from wikiquote:
The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.
Everything should be made as simple as possible, but no simpler.)
Arcfrk 07:20, 2 June 2007 (UTC)

[edit] Differential Equations and their Solutions

In my opinion there should be a heading similar to the one above. Below the heading would be, say sub-headings, the first one being, "first order linear". The general form of the equation being displayed and a wiki link to that title which gives it's solution, and so on for other known differential equations. But, I suppose we all have that same thing in mind, it's just a matter of someone having the time to do it. 130.36.62.140 16:04, 8 June 2007 (UTC)

[edit] Solving DEs?

A new section entitled solving DEs has popped up. I think there seem to be some inaccuracies in this, however i am not an expert on DEs, just use them. Anyway, my understanding is such that i think there are a few errors in the text as it stands. Firstly i didn't think that separation of variables worked on PDEs, nor does the characteristic equation allow for solving a linear DE of any order (analytically). I assert that you can't solve one of order higher than 4 analytically as this would violate the Abel-Ruffini theorem. Finally i don't think that it belongs in this section other than a statement that says something along the lines of "There is no method for solving every DE analytically, however some subclasses can be solved in this manner" Or something like that anyway. Thanks User A1 11:16, 27 June 2007 (UTC)

Thanks for pointing it out. Yes, the section is (perhaps unsalvageably) flawed. I have reproduced it here, because it does contain some information which may be incorporated into the article at a later point. Silly rabbit 11:33, 27 June 2007 (UTC)
Some solution methods can be found at Ordinary differential equations and Examples of differential equations. Abel-Ruffini theorem is irrelevant, since the meaning of 'an ordinary differential equation that can be solved analytically' (i.e. in quadratures) is completely different from the meaning of 'an algebraic equation that can be solved in radicals'. Arcfrk 19:44, 27 June 2007 (UTC)
I am afraid I don't understand what you mean, I just cant see how you can solve an ODE using by constructing a polynomial of order equal to that of the ODE (say quintic or higher) and solve the polynomial (analytically at least), which is why I invoked the A-R theorem, as suggested in the section entitled "The auxiliary equation method". If you have the time to explain it, thats great, but perhaps my talk page is most appropriate for this informative discussion. Thanks User A1 04:13, 28 June 2007 (UTC)

==Solving differential equations==

The method used is dependent on various characteristics of the equation previusly mentioned (i.e.order). They are used to find an equation in terms of the variables (typically y and x) and a constant c.

1. Separation of variables. This used to solve first order differential equations in the below form:

dy/dx = f(x).g(y)

In simple terms, it can be solved by dividing through by g(y) and multiplying by dx. The resulting equation can then be integrated on both sides to remove dx and dy terms.

2.The auxiliary equation method This can be used for an equation of any order. It works by simply replacing the dy/ dx term by a constant, lambda. d2y/dx2 is replaced by lambda squared etc. The resulting equation is then solved to find lambda. For non-homogeneous equations see below heading. The solution is the put into the below form. Let lamda take on two values in this case. One being equal to c, the other equal to d.

y= Aecx + Bedx Where A and B are constants to be found by substituting in initial conditions.

3. Non homogeneous equations. Where the differential equation is equal to a particular expression. The particualr integral must be found.

Types of differential equations

   * An ordinary differential equation (ODE) is a differential equation in which the unknown function is a function of a single independent variable.
   * A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and their partial derivatives.
   * A delay differential equation (DDE) is a differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
   * A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
   * A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

[edit] "deterministic" link

not sure exactly where the link "deterministic" in the introduction should be pointing to, but probably not to a page on the determinism philosophy. can anyone clear that up? -- anon

A possible target would be Deterministic system (mathematics), but I'm not so sure that's helpful. So I simply removed the link. -- Jitse Niesen (talk) 04:51, 24 July 2007 (UTC)
Actually, I now see that User A1 had already decided that Deterministic system (mathematics) would be a proper target and changed the link to point to that page. Hence I'll leave the link. -- Jitse Niesen (talk) 04:53, 24 July 2007 (UTC)

[edit] Confusion

"In many cases, this differential equation may be solved, yielding the law of motion." -End of paragraph 1 of introduction.

There are three laws of motion. Which law are you referring to? Nschoem 00:45, 21 February 2008 (UTC)

You are confusing the explicit form of the function describing the motion ("the law of motion") with three Newton's Laws (which are laws of physics in the general sense; at most two of them concern motion; and none is called the law of motion). Arcfrk (talk) 07:39, 21 February 2008 (UTC)

[edit] Introduction

The introduction to this article is unnecessarily verbose. It currently reads:

Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities (modeled by functions) and their rates of change (expressed as derivatives) is known or postulated. This is well illustrated by classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's Laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved explicitly, yielding the law of motion.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

In some cases, the wording seems misleading. For example, the first sentence seems to imply that differential equations arise only when analyzing deterministic systems whereas stochastic differential equations exist and can be quite useful (in fact their article is linked to in this article). In addition, differential equations are used in quantum mechanics to model discrete systems.

There are a variety of words or constructions that seem to be inserted as hedges that are not fully justified and only serve to make the passage more obscure than it needs to be. What does it mean to say that they are "mathematically studied"? As opposed to being studied philosophically or historically? What does it mean that they are studied from several different perspectives (aren't most things?) or that most of these perspectives are concerned with their solutions? Are there studies of differential equations that are indifferent to their solutions, and if so why don't we move that information to a different section? As far as I can tell, all of the directions of study mentioned in the article are concerned with some aspect of their solution. The point of the second paragraph seems to be mostly that in practice differential equations are solved numerically rather than analytically. In that case, why not just say that and put other information (e.g. their non-mathematical study, or studies uninterested in solutions) in another part of the article?

Much better would be:

Differential equations relate a function and its derivatives. For example classical mechanics describes an object's motion by how its position and velocity change over time. We can relate the position, velocity, acceleration (among other forces) as a differential equation for the object's position as a function of time.

Solutions to differential equations are functions that make the equation hold true. Although many practically-important differential equations cannot be solved analytically, such equations can typically be solved using numerical techniques. In many cases, the qualitative properties of the differential equation can be determined without finding exact solutions.

SmartPatrol (talk) 19:28, 30 March 2008 (UTC)

You are absolutely right about "verbose"! This is because a part of the lead was moved to the first section, so basically, the same descriptions of the subject of differential equations have been duplicated. I've pondered for a while what to do it about it, but was not able to come up with a good solution. Please, have a stab at it. Arcfrk (talk) 18:07, 4 April 2008 (UTC)

[edit] Font size of equations

Is there any way to increase the font size of equations in the article? In particular, using WinXP + ClearType enabled + Firefox, it's very difficult to see that

u'

is, in fact, u prime. And I have better than 20/20 vision. Thanks, WalterGR (talk | contributions) 10:58, 12 April 2008 (UTC)

If the u' is inline in the text, I have changed this to u′. On a separate line, I have changed it to
u'\,

Is that better? silly rabbit (talk) 12:47, 12 April 2008 (UTC)

Much better! Thanks a bunch, WalterGR (talk | contributions) 12:56, 12 April 2008 (UTC)

[edit] Merger discuss

I added this section, since the main article referred me here, but there is no dedicated topic to the merger discussion.

I disagree with the idea of the merge, since I believe the "outside of physics" version can be less for geniuses, as someone noted above. There are reasons to merge them also, but I'm looking at this from utilitarianism. I think some people may click it, looking for less technical applications (e.g. high school students) Sentriclecub (talk) 13:45, 20 May 2008 (UTC)

  • Support. I support the merger, since the article Differential equations from outside physics is just a list — a very short and poorly maintained list. It serves no useful independent purpose and should be merged with the main article. (In fact, all or most of its contents are already here.) silly rabbit (talk) 13:56, 20 May 2008 (UTC)
  • Support per Silly rabbit.  --Lambiam 06:52, 22 May 2008 (UTC)

[edit] Order

Can there be a section on differences and examples of first, second, and third order equations? 165.134.208.22 (talk) 19:49, 5 June 2008 (UTC)