Talk:Ordinary differential equation

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[edit] Gripe with incorrection

Just a small gripe. The (Possibly complex) part of this is annoying me. It's not possibly complex, they'll always be complex.

I've changed "complex" to "imaginary," as that seems to be what was actually meant. You do realize that you could have just edited the article yourself? Ruakh 04:08, 12 May 2005 (UTC)
I did not mean "\in i\mathbb R". i meant "\in\mathbb C\smallsetminus\mathbb R". Kwantus 2005 June 30 00:14 (UTC)
Yes, he could simply edited it and get on with his life. OTOH, after he noticed the error, he decided to discuss it with the community. I don't see what's wrong with that approach. --Mecanismo 11:17, 3 December 2005 (UTC)

[edit] First paragrapho needs work

The first paragraph is not quite technically correct. It's not a bad start, but should be refined a bit.

If you could elaborate in its technical incorrectness, I'll get refining... GWO

Now you have something to pull apart...RoseParks

[edit] Typesetting note

Just a typesetting note: when doing some minor copyediting here, I discovered that using an ordinary apostrope (') for the "prime" sign really wreaks havoc on the Wiki software (especially when doubled), since it tries to interpret them as bold, italics, etc. The "correct" thing to do would be to use ′, but some browsers won't handle that well, and there's no other tricky notation on this page to justify the use of special characters. So I have used the bare-acute-accent character ´ (decimal 0180) for the prime symbol. It should work on all browsers using either ISO or Windows ANSI, and doesn't screw up the Wiki software. --LDC

Superb!

Now for the nits... (heh) For an ODE, we really speak of a function x of a single parameter t, that is, x = x(t)...

The really frustrating thing about math is that so much of the notation I learned in lower division I more or less had to unlearn in upper division or grad level classes.

[edit] Types of DEs and solving methods?

Perhaps some mention of the different types of differential equations and methods for solving them would be appropriate (i.e. linear first and second order, etc.) --BlackGriffen

[edit] Relationship to Vector fields talk?

No talk at all about the relationship to Vector fields, which are really ODEs wearing funny hats, except they exist on differential manifolds and can be defined without co-ordinates.

Would you like to add a paragraph about that connection? AxelBoldt

[edit] Plural?

Is there any reason why this article uses the plural title?

nope. should be fixed. AxelBoldt
yep. "Differential Equations" is the name of a field of study in math. A "differential equation" (singular) would be the object of that field of study. I will move the article to the singular, leaving a link from the plural. Ed Poor

[edit] History section needs rewrite

I believe that the history section here ought to be rewritten preferrably by someone who knows something about it and doesn't simply 'copy' a 100+ year old book on the subject - and doesn't tell that's what have been done! I see that there are a number of references to 'recent works', 'the modern school' and so on, when referring to texts written well over a century ago! 'Recent writers' refer to (amongst others) Klein (-1925), Weierstrass (-1897) and Frobenius (1849-1917). Do you agree that the history part here should be deleted? Mikez 18:00, 23 Feb 2004 (UTC)

Please don't delete. Quite a bit of work has been done already - obviously it is still in a bad state, but starting again with nothing isn't a good idea. In the end there will have to be major changes - of course.

Charles Matthews 18:13, 23 Feb 2004 (UTC)

This now removed - hard to upgrade:

The modern school has also turned its attention to the theory of differential invariants, one of fundamental importance and one which Lie has made prominent. With this theory are associated the names of Cayley, Cockle, Sylvester, Forsyth, Laguerre, and Halphen. Recent writers have shown the same tendency noticeable in the work of Monge and Cauchy, the tendency to separate into two schools, the one inclining to use the geometric diagram, and represented by Schwarz, Klein, and Goursat, the other adhering to pure analysis, of which Weierstrass, Fuchs, and Frobenius are types. The work of Fuchs and the theory of elementary divisors have formed the basis of a late work by Sauvage (1895). Poincar\'e's recent contributions are also very notable. His theory of Fuchsian equations (also investigated by Klein) is connected with the general theory. He has also brought the whole subject into close relations with the theory of functions. Appell has recently contributed to the theory of linear differential equations transformable into themselves by change of the function and the variable. Helge von Koch has written on infinite determinants and linear differential equations. Picard has undertaken the generalization of the work of Fuchs and Poincar\'e in the case of differential equations of the second order. Fabry (1885) has generalized the normal integrals of Thomé, integrals which Poincar\'e has called "intégrales anormales," and which Picard has recently studied. Riquier treated the question of the existence of integrals in any differential system and gave a brief summary of the history to 1895. The later contributors include Brioschi, Königsberger, Peano, Graf, Hamburger, Graindorge, Schläfli, Glaisher, Lommel, Gilbert, Fabry, Craig, and Autonne.

Charles Matthews 11:52, 12 Apr 2004 (UTC)

Ok...Mikez

[edit] L-H revert

Unfortunately, I can't make sense out of the sentence "Differential equation was born as the fundamental equation which describes the natural law." It contains grammatical errors which could be fixed, but the meaning is still too unclear to me. So I reverted. I notice that the sentence in question links to fundamental equation. I question whether this usage is standard.

Seems like a lot of the history on this page may be a copyvio? We definitely at least need the source of all this stuff. I'm working on it to make sure the dead men don't seem to be alive  :-) - Gauge 21:18, 4 Aug 2004 (UTC)

No, from an old PD source I believe: User:Recentchanges added this and similar stuff on a number of pages. Charles Matthews 06:15, 23 Aug 2004 (UTC)

How much schooling do I have to go through in order to be able to understand ANY of this? I am just curious: what is the education level of the authors of this page?

With nothing more than differential calculus you can understand what differential equations are and what it means to solve a differential equation. To get much into the theory of the subject, you need to go beyond first-year calculus. Michael Hardy 00:57, 1 Sep 2004 (UTC)
As to education level of authors of this page, I suspect it varies greatly, since differential equations is (yes "is", not "are") one of those courses that very large numbers of students in many different fiedls are required to study. Generally that means lots of people who don't know much math have contributed here, probably including some who haven't gone beyond a couple of years of calculus. Michael Hardy 14:51, 1 Sep 2004 (UTC)

[edit] Splitting up the page

I was thinking about reworking this page, and breaking out a lot of the topics into their own pages (like ordinary differential equation, method of undetermined coefficients, etc.). Does anyone object? -- Walt Pohl 07:08, 7 Jan 2005 (UTC)

I don't object conceptually - indeed, something much like that is on my to-do list - but I do advise caution, as many articles already exist on subtopics (for example, I came across variation of parameters after completely reworking the section on it in this article), and content from this article needs to be merged into those articles. The last thing we need is even more articles on subtopics duplicating each other. Ruakh 06:55, 8 Jan 2005 (UTC)
One thing I think should be done is get rid of the separate examples of differential equations article and move its content to this article and/or articles on subtopics. Ruakh 06:55, 8 Jan 2005 (UTC)
You're right. That information should be integrated better. And thanks for pointing out the variation of parameters page -- I doubt I ever would have found it myself. -- Walt Pohl 09:06, 10 Jan 2005 (UTC)

Okay, I've formulated a possible (albeit complicated) plan of attack:

-- Walt Pohl 09:27, 10 Jan 2005 (UTC)

The new problem is, of course, that the new method pages require some background before getting into them. method of undetermined coefficients, for example. I'm working on it, but there's no real easy solution. It's hard to provide context for these method pages without slogging through the whole ODE mess --Eienmaru 23:14, 15 May 2005 (UTC)

[edit] Copyright violation?

Moved from article:

This seems like probable copyright violation; what is "3.1"? Ans: "3.1: Linear ODEs with constant coefficients -=Heuwitt=-" A section of a non-public-domain work? Ans: "You are welcome to contribute additional examples (which pertains to the topic of interest, that is). -=Heuwitt=-" Ruakh 21:03, 2 Jan 2005 (UTC)]

I don't agree that the use of "3.1" may indicate a probably copyright violation; Heuwitt's explanation seems reasonable to me (though I would refer to the section in another way). -- Jitse Niesen 13:54, 19 Jan 2005 (UTC)

[edit] eh... beh?

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. I'm sorry but I have no idea what this paragraph is trying to say, or what it has to do with the section it's heading. Anyone care to rewrite it? PenguiN42 21:21, Jan 19, 2005 (UTC)

[edit] Euler's Forumla

Is it truly necessary to derive Euler's formula in the middle of this article? Can't we just take it 'on faith' and link the astute reader elsewhere? --Eienmaru 02:55, 22 May 2005 (UTC)

I agree. That just distracts from the main point of the article. I now replaced it with a link. This article looks as if it needs more work. Oleg Alexandrov 03:36, 22 May 2005 (UTC)
IMO even mentioning Euler's formula clutters things up. So does an example that switches to operator notation. It's making a really really simple case into a hash. Kwantus 2005 June 30 00:39 (UTC)

[edit] Homogeneous Linear ODE's: Error in example?

In the example side box, it says:

 e^ix, e^-ix, e^x, xe^x 
 This corresponds to the real-valued solution basis
 cosx, sinx, e^x, xe^x

Should this instead read cosx, cosx, e^x, xe^x

as cos(x) = cos(-x)

I don't think so: both cos(x) and sin(x) are real-valued functions that are (complex) linear combinations of eix and e−ix. Michael Hardy 22:12, 28 August 2005 (UTC)
To elaborate on what Michael Hardy said: it's not that eix corresponds to cos(x) and e−ix corresponds to sin(x); rather, eix and e−ix, taken together, correspond to cos(x) and sin(x). Does that make sense? Ruakh 02:00, 29 August 2005 (UTC)

well, actually (a far as i know...) there is a formula saying that : e^(i*T) = cos(T)+ sin(T) ; where T is a mathematical expression Undye 13:23, 25 January 2006 (UTC)

You're close. According to Euler's formula, e^{iT} = \cos \left( T \right) + i \sin \left( T \right). I don't quite see your point, though . . . ? Ruakh 15:00, 25 January 2006 (UTC)
Hey Undye, I see from your entries that you're a new wikipedian. Welcome! Here are a couple of hints: When you respond on a talk page, use colons (:)s to indent the conversation. Also, you can sign and date your entry by putting 4 consecutive tildes (~) at the end of your entry. Please do so, so we can get to know you. As to your question, as Ruakh and M.Hardy said: Two of the solutions are complex-valued functions (e with an i exponent). Using Euler's formula, you can transform the complex functions to real-valued functions. Although people discuss ODEs in terms of calculus, I think you can get more intuition for this type of transformation from linear algebra. The idea is that your solution is composed of several functions that can be added up (linearly combined) to the complete solution. You call these functions the basis for your solution. Some of these functions might have characteristics that you don't like (such as functions that have complex values). So you can transform them to other basis functions. [edit: after all that I forgot to sign!!!] Tristanreid 17:31, 25 January 2006 (UTC)

[edit] Characteristic equation?

In the section Homogeneous Linear ODEs with constant coefficients, it defines F(z) as the characteristic equation of the differential equation. Can someone explain how that jump makes sense? Why is it that solving the roots of this polynomial magically points us towards an answer? —BenFrantzDale 23:47, September 12, 2005 (UTC)

Basically, since you know that exponentials have the property that their derivatives are nearly the same, you can just take an educated guess that some form of exponential is the solution to the diff eq. Plug the guess in to the diff eq, and you will see that the guess was right, but only if the exponential has a root of the characteristic polynomial in it. For example, suppose I want to solve y’’-3y’+2y=0. I guess that the solution is an exponential, like y=erx, where r is some unknown coefficient. If this is to be a solution, then I must have r2erx-3rerx+2erx=0, or (r2-3r+2)erx=0. Since erx≠0, we have (r2-3r+2)=0. Thus, r must be a root of the characteristic polynomial. -Lethe | Talk 20:24, 13 September 2005 (UTC)
Aah. Thanks for clarifying. I may make some note of that in the article. About this "educated guessing", can you show that the set of these exponentials provides a basis for all solutions? Clearly the method works, but I'm not finding a proof that all solutions are of that form. Thanks. —BenFrantzDale 22:15, 13 September 2005 (UTC)
In fact, it won't get all of them, if there is a repeated root. Then you should try a guess like xerx, or higher powers if the root is repeated more often. Once you have n independent solutions to an n-th order diff eq, you can be sure you have all of them. -Lethe | Talk 22:38, 13 September 2005 (UTC)
I edited the page accordingly. I think it's pretty-much correct, and more clear to the uninitiated (or at least the rusty). Feel free to give it more polish, of course. —BenFrantzDale 23:23, 14 September 2005 (UTC)

[edit] Existance and nature

I just added an "existance and nature" section to the page. It could probably use a review by an diffeq expert.

One thing in particular, I coined the term "hybrid solution" to refer to a solution assembled piecewise from particular and singular solutions, though I've never seen that term used in literature. Any suggestions for a more standardized wording?

Baccala@freesoft.org 23:50, 29 October 2005 (UTC)

[edit] Please respect the editorial style of a discussion section

I've seen a lot of comments being added which ended up being a PitA to follow through or even understand. That happens because the commenter simply didn't followed the basic rules of adding comment. Let me mention a few of them.

  • If you start a fresh comment, write it in a different section.
  • Make it clear where the comment starts and where the comment ends (section, indentation, etc...)
  • Always. sign. your. comment!

There are other rules but these are the basic, have the biggest impact and were the ones which were lacking the most. If you are going to write a comment, please obbey them. It makes the life of who's reading the comments a lot easier and let's your voice be heard clearer. --Mecanismo 11:29, 3 December 2005 (UTC)

[edit] Vote for new external link

Here's my site full of ODE example problems. Someone please put it in the external links if you think it's helpful!

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

[edit] please tell me

how to solve y'+cosxy=1/2sin2x —The preceding unsigned comment was added by HydrogenSu (talk • contribs) .

Meaning y'+(cosx)y=1/2(sin2)x or y'+cos(xy)=1/(2sin(2x)) or perhaps y'+(cosx)y=(1/2)sin(2x) ? Bo Jacoby 12:03, 27 January 2006 (UTC)

My guess is y'+cos(xy) = (1/2) sin(2x). Prime denote derivative wrt x. --Salix alba (talk) 12:19, 27 January 2006 (UTC)

The unknown function y is rarely argument to a cos, so my guess is the linear nonhomogenous equation y'+(cosx)y=(1/2)sin(2x), but we need to know. Bo Jacoby 12:31, 27 January 2006 (UTC)
Sorry but previous writting was not clear. Should be
y'+[cos(x)]*y=\frac{1}{2}sin(2x)
Thank you two gentlemen. —The preceding unsigned comment was added by HydrogenSu (talk • contribs) .

The formula of the article Ordinary differential equation#General solution method for first-order linear ODEs gives

y=e^{- \sin x}\left(\int{\frac{1}{2}\sin (2x) e^{\sin x}dx} + C \right)

where C is an integration constant to be determined from the initial conditions. The integral remains to be reduced. Bo Jacoby 14:16, 27 January 2006 (UTC)

Please don't write

y'+[cos(x)]*y=\frac{1}{2}sin(2x).

Instead, write

y'+[\cos(x)]\cdot y=\frac{1}{2}\sin(2x).

This not only de-italicizes "sin" and "cos" but also causes standard spacing conventions to be observed. Also, note that I did not use an asterisk to represent multiplication. Michael Hardy 22:56, 27 January 2006 (UTC)

[edit] Bibliography: Self-references

Here, and mainly in many other math-related articles, User:Rea5, and other anonymous IPs (probably a dynamic IP) have been adding references to a book authored by Refaat El Ataar. This is not a notable math book (specially because it was edited in 2006!), so many users have been reverting those reference inclusions. Probably, it's a self-reference. (this may be coincidence but the user name Rea coincides with the initials of the author).

If you are the user who includes this references, please discuss it here first and explain why you think that book should be listed here. Otherwise, references to Refaat El Ataar books in this article will keep being removed.

--John C PI 14:40, 31 January 2006 (UTC)

[edit] Relationship with Wikibooks

As a newcomer to the editing business, I'm confused about the relationship between an article on Differential Equations and the Wiki textbook on the subject. Looking at the ODE article as it stands, it appears that much of the material belongs in more systematic treatment, as in a textbook. What should appear in the main articel should be basic definitions, some explanation about why the topic is interesting or important, and references to further reading, such as an on-line textbook. With that in mind, I would propose to add some elementary material explaining some applications and maybe some simple techniques for obtaining solutions. The more intricate stuff should be stuck in the textbook. Donludwig 17:28, 23 February 2006 (UTC).

I agree, this page is unneccessarily huge - and largely unreadable. I came here looking for a quick reference for second order DEs, but found no help from this page. Fresheneesz 07:15, 14 April 2006 (UTC)

[edit] What are PDEs doing here?

Why is there a section on PDEs? The page title is ODEs! Please can someone who knows how to do this move the section on the history of PDEs to the PDEs page. (Or delete it entirely - it's not very good.)

[edit] Forcing fuction

The FF dab page points here, but I can't see any FF info. Rich Farmbrough 12:23 26 June 2006 (GMT).

[edit] Method of undetermined coefficients vs. method of annihilators

I noticed that the article on the Annihilator method simply redirects here. However, there doesn't seem to be enough of a discussion about annihilators on this page to justify that. Specifically, this page treats the method of undetermined coefficients and the method of annihilators as being synonymous, which I don't think is exactly the case. The method of annihilators can be used as part of undetermined coefficients, but I think they are separate processes. Indeed, the article on Method of undetermined coefficients doesn't even mention annihilators, and the examples on that page are accomplished differently than those given here. Unfortunately, I just don't know enough about the method of annihilators to feel comfortable changing the reference on this page, and maybe creating a bigger article for it. Does anyone have more information about the relationship between these two methods? Thank you, Rundquist 23:13, 1 September 2006 (UTC)

[edit] The derivative notation

In Part #1 of the introduction, the left side of the decay equation is worded as "the derivative of u, divided by the derivative of t", considering that the dx/dy form is just a formal notation and does not imply actual division shouldn`t it say "the derivative of u [as function of t]" ?

Yes. This is explicitly wrong. I will fix it for now, perhaps if someone can defend the previous revision, we will consider switching back. 48v 20:01, 8 September 2006 (UTC)

[edit] A little pedantic

Just above the section titled "Mathematical Definition" I feel the last line of the y equation should say if x \geq 2, rather than having focus on two, I just think people are used to the notation I'm suggesting and might accidently read it wrongly (like I did).--Ultimâ 10:30, 19 September 2006 (UTC)

[edit] Intro simplified

This page was in a bad state, especially the introductory section which was full of advanced concepts and a complicated example. I have simplified it down to a basic introduction. I have also tried to improve the structure further down (up to first-order linear) but below there it is still a mess, for example variation of parameters is described twice. Paul Matthews 13:01, 25 October 2006 (UTC)

[edit] Scalar vs Vector ODEs

The article seems to deal only with scalar odes; INHO, it would be advantagous to stress the similarity with the vectorial case, and state in particular that higher-order odes can be recast as vector odes. --Benjamin.friedrich 10:33, 9 November 2006 (UTC)

[edit] Make article accessible to readers outside mathematics

When I read the article I personally do have the impression that it is a colorful mix of different topics; leaving the reader a bit confused about what an odes actually is. All of us visiting this discussion page probably know what an odes mathematically is, but how to define it as a concept of thinking? I think answering this question is important to make the article accessible for someone outside the quantitative sciences. The answer to this question determines the style of the article. A provisional list of answers could be

  • an equation one can do interesting math for
  • a vital tool to do physics
  • a historically important branch of mathematics
  • a clever way to define flows in state-space

The classification of ODEs and ways how to solve them now taking up much of the article in nicely written, but I am contemplating whether one should make them an extra article.

In my opinion it would be benificial for readers to stress right in the beginning some practical facts about ODEs such as

  • in most cases they have a unique solution whenever initial conditions are specified
  • in most cases it is impossible to find this solution analytically although centuries have been spent to solve odes
  • geometric thinking helps a lot when you deal with odes: often the independent variable is time, the dependent variable is the state of your mathematical object or physical system, and then the ode will tell you how your system will evolve
  • numerical integration is a standard procedure
  • the theory of odes was started by Newton, allowed impressively to predict planetary orbits and I dare to say boosted enlightning.

Please let us discuss these issues. --Benjamin.friedrich 10:58, 9 November 2006 (UTC)

I completely agree, it reads more like a textbook than an encyclopedic article. I will try to rewrite the article and put the focus on the core ideas instead of the examples. MathMartin 15:40, 1 December 2006 (UTC)

[edit] Rewrite as of 11.12.2006

The article is currently in a very bad state. In order to add the material mentioned above we first have to clean up the article:

  • remove most of the worked out examples
  • move content not directly relevant to other articles
  • add core ideas and methods

I did some heavy editing today in order to accomplish these goals, but now I am tired. I will resume working on the article in the next few days so bear with me for a while. MathMartin 20:59, 11 December 2006 (UTC)

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