Talk:Linear differential equation

7523:. The linear differential operator consists of a sum of derivation operators of any degree, each single operator multiplied by a coefficient function. Since the simple derivative operator is linear, its composition (second and higher derivatives) and also any linear combination of these is also linear (there is no product of derivatives, or any other non-linearity). As a consequence of the linearity of the operator solutions to the LDEQ can be added together in particular linear combinations to form further solutions. The key goal in analyzing any differential equation is to find the general solution, a family of functions 95: 85: 64: 31: 8114:, we find much material on the characteristic polynomial of a square matrix for computing eigenvalues, a brief mention of the characteristic polynomial of a graph (adjacency matrix), but no mention of the characteristic polynomial of a linear differential equation with constant coefficients. This inconsistency can be confusing for students who have just begun studying this topic. 22: 7473:- The second sentence about the "value of a particular type of polynomial" is to my measures (a)fully indigestible, and (b)induces wrong associations wrt to "degree" of an derivative, and of an (multiplicative) exponent. The algebraic coincidence of the linearity-disturbing "multiplication" and the linearity-conserving operator "composition" has to be cautiously regarded. 8262:(i) We edit this article to explain how the characteristic equation associated with a differential equation is related to the characteristic equation associated with a matrix (à la Boyce and DiPrima), thereby justifying the terminological similarity; (ii) we create a disambiguation page "Characteristic polynomial" (in the same vein as the disambiguation page 4359:. The exponential function is one of the few functions that keep its shape even after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. Thus, to solve 7248:

signal processing and physics, the concepts such as "amplitude of the input", "angular frequency of the input", "amplitude of response", "gain", "phase lag" and "complex gain". This is the difference with other articles similar to ours. We think that our articles are of practical value, we hope our article will post on the Knowledge.

5308:. The exponential function is one of the few functions that keep its shape even after differentiation. In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function. 5384:). I'm unfamiliar with that notation, the edit doesn't explain, and a couple of other ODE solutions using Cramer's rule/Wronskians don't seem to include it. However, not being an expert, it would be great if someone more familiar with the subject could take a look at it (and maybe clarify). Thank you very much! 7840:

This is much better. There had been complaints in the talk page above going back to 2013 about the statement that solutions could be added together to form other solutions. My only comment now relates to the phrase "if the constant coefficient b(x) is zero" at the end of the lead. In general, b(x)

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Associating the simple derivative operator, which yields the derivative when applied to a function, and its various degrees with the corresponding powers of the indeterminate in a polynomial (the zero-th power of the derivative operator being the identity), and the coefficient functions, multiplying

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The article is currently inconsistent with itself regarding whether a linear differential equation is allowed to be inhomogeneous. I think the common usage is to allow this, and to refer to homogeneous linear differential equations when there is no inhomogeneous term. I would advocate changing the

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With the current title, users might misunderstand this article and think that it applies to differential equations in general. In fact, I personally know of one person running around the internet who tried applying this article to show that Maxwell's equations give unphysical solutions, because they

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To me, this article seems to have a pure mathematical focus on the existence of solutions. The definition of a linear differential equation should also be relevant to applied mathematics, where it is quite common for the coefficients and especially the forcing term, b(x), to be discontinuous. For

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We took this as a problem and deepened the research and decided to post the generalized ERF method to the cite. We have described the generalized ERF method and practical examples in the article. This is clearly an improvement on the previous article. This is the first reason our articles should be

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understand why you two want to rid it off. ERF is important concept on which was nothing here before I made the modest contribution. Existence of the page has no harm, but deletion the page will return a hole. I believe having everything on a single long long page is not good option. I am expanding

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As Sandrobt points out in the "vector space?" section above, it is not true that "linear differential equations solutions which can be added together to form other solutions" or that "the solutions to linear equations form a vector space". These comments should be moved from the article lead into

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The contents of this article are pretty much the limit of my knowledge of differential equations; in particular, I don't know anything about Picard–Vessiot theory or differential Galois theory. Perhaps it's important that the coefficients of higher-order differential equations be differentiable.

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Considering that I tried to save as much as possible of the current content (so most of the last two paragraphs), and my efforts to just remove obvious flaws, it is in my opinion sensible to dump the current lead, and that interested editors start to improve on my reinstated suggestion and make it

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I have made some significant notation changes. It is very important that consistent math notation be used in a article. There are different conventions. In this article, I am focusing on the use of y and t, rather than y and x for elementary linear differential equations for the simple reason that

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Sorry that I'm a slow respondent! I generally agree with not making things more complicated just because one can, especially in an encyclopedia article. In this case, however, I'd simply want to replace one technical term (differentiable) with another (continuous). But I want to make sure that

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You are right. However, in practice, the coefficients are generally differentiable, and even analytic. Giving in each case the weakest conditions for making the results true would make the article more technical, without adding much useful content. This is the reason for which I have chosen to be

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IMO, the interpretation in terms of differential operators is too technical for appearing in the section about basics (and it is not needed for explaining the basics). It should be moved below, maybe in a section "Linear differential operator". This would have the advantage of allowing a clearer

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The alternative equation using the delta-Dirac function looks questionable to me. The limits on the integral are a and x. The variable a should be dimensionless while x has units of the independent variable. Because they are not commensurable, I do not see how they can appear here. Am I correct?

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The main conclusion from this sample is that there is no wide agreement between authors, so the decision whether to use "characteristic equation" and/or "characteristic polynomial" in the context of linear differential equations with constant coefficients is left to the editors of this article.

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I think that all the intellectual property that human beings make is shared by web cites whether it is large or small, and is contributed to the intellectual development of mankind. We have posted examples and applications with the motivation for this formula to be called "ERF". For example, in

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From the German language article on the subject we have the following list (all of which I am sure are also listed somewhere in the English Knowledge. Birkoff and Rota introduce the subject of second order linear differential equations with the Bessel differential equation (number 2 in the list

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The introductory paragraph struck me as rather misleading, as it implied that any linear combinations of solutions to a linear differential equation are also solutions to the equation. I've slightly reworded it to make it clear that this property is only true when the equation is homogeneous.

7774:(PDEs). An other property to categorize LDEQs is the highest degree of a derivative operator (the "degree" of the formal polynomial). This number is called the degree of the DEQ. Restricting the coefficient functions of the derivative operators to constants yields the class of LDEQ with 7224:. This article outlines linear differential equations and gives a representative solving method. However, you did not give specific information about ERF. The formula given in this article simply outlines the ERF formula when P(r) is nonzero. Thus, the next question comes naturally. 4699:

Secondly, I have a problem with the statement "The exponential function is one of the few functions that keep its shape even after differentiation." Is it not the case that the exponential function is uniquely defined by this invariant property? The new text will be "Thus, to solve

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Also, I believe the full citation should appear in the References section and the author, Mário N. Berberan-Santos, credited in the reference: Berberan-Santos, M. N. (2010). "Green’s function method and the first-order linear differential equation." J Math Chem, 48(2), 175-178.

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should be avoided. The ERF concept seems "independent" enough to have individual article, as far as I am concerned. However, further work on the article is needed, mainly in rephrasing and formatting of the current content and adding new citations to support the claims.

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The next task is to transform each of the above into a "standard" notation such as the "D" notation; the most sensible place in which to record this is in the above list of equations to see how they look. The list is more or less complete now. I have pedantically used

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In several places in this article, it is stated that the coefficient functions in an ordinary linear differential equation must themselves be differentiable. However, it seems to me that everything in this article makes sense as long as the coefficient functions are

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I don't think this is right. The three solutions should not be strung together separated by equal signs, they're not equal. In particular, the exponential form is the most general solution, while the sine and cosine forms are more limited possible solutions.

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legitimate content then at the source article; as I cannot confirm that the source references support anything, it consider it likely that some suggest the content I provided. The "background" in the source article is really not appropriate for Knowledge. ?

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As an apology for my inept attempt on a new lead I want to explain that I tried to cling as much as possible to the current content of the article, thereby feeling bound to the polynomial view on the operator, instead of on the LDEQ as a whole. Bests.

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I have also removed the section "Exponential response formula", and replaced it by the explanation of the cases where this method and other related methods apply. This seems me the best way for making the article useful for the layman (see

598: 4839: 4498: 8019:. In particular, the one place where details about solutions are given without additional restrictions on the coefficient functions (for first-order equations), the solution shown is correct for any continuous coefficient functions. 8259:(i) We edit this article so as to use only the term "characteristic equation" in the context of differential equations; and (ii) we don't create a disambiguation page "Characteristic polynomial" because there is no confusion anymore. 1543: 6940: 8241:

Another conclusion is that a majority of authors seem to use "characteristic equation" more often than "characteristic polynomial" for differential equations, but it is hard to tell whether my sample is representative or not.

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Ly = f where the differential operator L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side f is a given function of the same nature as y (called the source term).

6416: 2773: 7867:- I think the linear superposition of homogeneous solutions to inhomogeneous solutions, vaguely used for defining LDEQs in the previous lead, should get more emphasis than just touching it en passant with vector spaces. 7448:

I believe that the proposed version of the lead is at a level that is way too advanced for many readers, including those who at the stage in college when the are deciding whether to take a differential equations class.

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It is also the case that the "dot" notation for differentiation with respect to time features widely in the literature. I will try to present it in appropriate contexts (with modern up to date supporting literature). —

6707: 6269: 2052: 7485:- Talking about "terms" within polynomials (which lack any sound basis here) instead of discriminating abstract "powers" of the indeterminate and their coefficients is not sufficiently precise in this here context. 5001: 4660: 8266:); (iii) we rename the article "Characteristic polynomial" to "Characteristic polynomial (matrix)"; and (iv) we move the few sentences about the characteristic polynomial of a graph to the new disambiguation page. 8256:); (iii) we rename the article "Characteristic polynomial" to "Characteristic polynomial (matrix)"; and (iv) we move the few sentences about the characteristic polynomial of a graph to the new disambiguation page. 7960:. They belongs to this article because they are the solutions of linear differential equations with polynomial coefficients. Although relatively recent (1990), this theory seems absolutely fundamental by unifying 1409: 7464:

Thank you for not leaving it to me to open the discussion: I would not have done it, because I am convinced that this article is outside an effective reach of my abilities. About my rewrite and the reasons for

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the page, it's going to be as long as LDE page is. Also lets name everything by its name, what you two offer is not merge but deletion. I hope Knowledge society is more tolerant to newcomers than you two are.

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I will check each of these in the English Knowledge (and references) and consider how thay might be written in a uniform way in the style for the article under consideration (using D notation, for example). —

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The technical information is good, but it doesn't really explain in plain english what a linear diff. e.q. is. It probably needs a section explain it in general terms, as opposed to mathmatical terminology.

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is a practical webcite to linear differential equations. However, the theory isn't deep. I think that's because of the vast theory and method of "linear differential equations". This is not author's fault.

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It remains certainly many typos and grammar errors, as well as other needed improvements. I may also have omitted some fundamental aspects of the subject. Be free of improving this fundamental article.

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explanation of the reason why the solutions of a homogeneous equation form an vector space, and why the general solution has the structure of the inverse image of an element under a linear application.

8860:, the general solution formula for first-order linear differential equations. I had not heard of the name and I am unsure as to whether it is actually used in English, but the remaining content of 7408:

Friends, propose to hold on with negative critiques and have a look on results. I and Phillip really want to make an A+ article. Any help with it as well as positive critique is extremely valuable.

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form really is excellent after all. In an old book by Birkoff and Rota, Ordinary Differential Equations (3rd edition 1978), I note how they tried to cope with this problem. They used the classical

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the single operators in the linear differential operator with the polynomial coefficients, this linear differential operator can be considered as a formal polynomial. Denoting this polynomial with

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merge. My name is Phillip and I am a mathematician, a co-author with Wandalen. I am going to leave some of my words for discussing in here. I would really appreciate it if you read this carefully.

5224: 3783: 3465: 7871:- I am unsure how to connect the polynomial view of the lead to the concept of the linear differential operator, used in the 1. section, and to the characteristic polynomial in the 2. paragraph. 6514: 8477: 3315: 151: 8922:. I would suggest that one of these is enough. Since I could find no evidence at all of the term "Armour formula" being used for this in English, I will propose deleting that article. I am 8792:

antiderivative. So the integral does not include the constant of integration. Otherwise, as there are two integrals in the formula, one would have a solution that depends on two constants.

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For the third point, this has been cited only as an example. I agree that more should be said about the content of the article, but this is difficult to do this before cleaning the body up

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And I hope, (not only) newcomers get more tolerant to opposing opinions (than you currently write). I won't spoil the !voting with explicating my opinions, but argue them on the ERF-talk.

3684: 783: 7907:, which are the solutions of linear differential equations with polynomial coefficients. In fact, most usual functions are holonomic, and this allows to automatize most of calculus (see 5433:

The incipit contains the sentence "The solutions to linear equations form a vector space", which is not really correct. This is true only for homogeneous Linear differential equations.--

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of this derivative to be non zero, it is eliminated by dividing through it. In case it can become zero, different cases must be considered separately for the analysis of the equation.)

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I have done a little rewriting of the introduction to make it more accessible. And I have added a simple example on radioactive decay taken from the book by Robinson 2004. He uses the

8545: 8228:"characteristic equation" is used for differential equations only once on pp. 134; instead, the term "auxiliary equation" is used repeatedly on pp. 134, 135, 136, 138, 139, 141, etc. 7863:- The notion "linear polynomial in several indeterminates" is not really explicitly covered in the given link, and is possibly not immediately accessible to the intended readership. 6124: 2467: 2314: 5609: 7353:

since 02.06.2017, which did not change the already mentioned verdict of there being "less here than meets the eye", evidently in response to obstruct any efforts to implement the

6052: 1141: 7968:, and making algorithmic many operations of calculus (such as antiderivative), for which there was only heuristics with the standard definition and representation of functions. 6571: 6135:

of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular, the following solutions can be constructed

7567: 8084: 2156: 1670: 1272: 1194: 3211: 2408: 8292:" at the end. I have thus edited the article for making this clear here. My opinion is that a similar edit must be done (if needed) in the matrix article and the dab page. 5520: 7388:

Sorry, did you call my and Phillip's contributions obstruction and pseudo-activity? Purgy, sorry, it sounds very offensive. What's wrong in desire to make a good article?

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such equations try to capture processes over time. Currently, in the article, the exponential solution for the homogeneous equation is introduced with respect to z and x.

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For theoretical reasons in constructing the complete family of solutions for the above LDEQ the associated homogeneous LDEQ with the same differential operator but with

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The first point may be solved by defining first the ordinary case for putting the displayed formula immediately after the link, and defining the partial case afterward.

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or higher. However, as the author of this major revision, I am misplaced for rating this article. So, please, review this article, and upgrade the rating as needed.

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These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:

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the link between the "characteristic equation" used for differential equations and the "characteristic equation" used for matrices is described on pp. 406-408.

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There is a typo in the formulation of the general solution which is expressed as y(x). In this case the integration variable in the integral is not t ...

6761:... a ... differential equation ... G(x, y, y', y") = 0 ... is said to be linear provided G is linear in the dependent variable ... and its derivatives ... 8993: 1420: 141: 6767: 7078:

The phrase "of the same nature" is unclear, and should be clarified -- as it stands, the reader is left to wonder what exactly it is that y and f share.

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slightly less general for, I hope, making the article easier to understand. This is a personal choice, which can be the subjectject of a discussion.

7826:. Thus I have written a new version, which (I hope) is correct, elementary, and contains all (and even a little more) material of the previous lead. 1793: 8973: 8252:(i) We leave this article as it is; (ii) we create a disambiguation page "Characteristic polynomial" (in the same vein as the disambiguation page 8023:

For this reason, I hesitate to simply change every appearance of differentiable coefficients into continuous coefficients. (And of course, the

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This clearly falls under the subject of mathematics, and even if it somehow is not then linear differential equations are still of that form...

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Therefore I would suggest to merge it to here unless significant new content on the formula is expected that is beyond the scope of this page.

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This article currently uses the term "characteristic equation" once and the term "characteristic polynomial" 11 times. But in the article

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it's correct before I do so! (I could start by making the replacement when I know that it's correct, which is the first-order case.) —

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to this article. I tried to integrate the material a bit but was not very successful. This article seriously needs a complete rewrite.

8911: 188: 5815:{\displaystyle y_{0'}={\tfrac {1}{2}}\left(A_{0}e^{ikx}+A_{1}e^{-ikx}\right)=C_{0}\cos \left({\tfrac {kx}{2i}}\right)=C_{1}\sin(kx).} 4258:

I am going to try to put some order on this article. First I will begin by adding at least one reference work which I currently use:

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Do you agree with me? Please read our article once again. You can find something dominant. I am waiting for your expected response.

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The next stage is to use uniform notation for all equations (where possible) and to cite sources (other than the Wikipedias). —

2612:{\displaystyle \ x(x-1)y''+\left((\alpha +\beta +1)x-\gamma \right)y'+\alpha \beta y=0,\ \alpha ,\beta ,\gamma \in \mathbb {R} } 8958: 8288:

Here as well as in the matrix case, a "caracteristic equation" and a "characteristic polynomial" differ only by having or not "

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In the gigantic literature on linear differential equations, I checked the use of these two terms in the following references:

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Even if it does, it will bring boredom of readers. Therefore, each parts are described in depth by its own cites. For example,

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notation as illustrated above. It is important precisely because it is already used in the examples later on in the article. —

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I must confess... that I agree with you. A linear differential equation is "just like" a line, but a line in general form. So!

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Avoiding confusing change of notation (passing many time from operator language, with several notations to equation language.

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which equate the result of applying a linear differential operator to some unknown function with some other function, called

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leads me to believe each solution example was to be labeled, but it's not obvious to me how the others were to be labeled. (

5381: 2475: 174: 7445:’s major revamp of the lead. Given the substantial extent of the reverted changes, the rewrite needs to be discussed here. 7040:

Potentially there may be a neater/more elegant way to write this though, as the paragraph is already quite bracket heavy.

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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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shows that, back in 2006, someone added "_{0 \choose f}" to the last equation before the example subsection in section

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Examples of differential equations#Non-separable (non-homogeneous) first-order linear ordinary differential equations

1678: 8131:"characteristic equation" is used repeatedly for differential equations on pp. 140-143, 145, 158, 160-163, 228, etc. 7941:

Removing examples that do not provide insight, or changing them for reflecting the section where there are displayed

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were unaware that this article only applies to ODE's, not PDE's. Otherwise I think the content looks pretty solid.

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Previous lead was a mess, and was difficult to understand even for experts. Purgy version is better, but much too

3224: 8763:{\displaystyle y=ce^{F}+e^{F}\int ge^{-F}dx=ce^{F}+e^{F}(H(x)+c_{1})=e^{F}(H(x)+c_{1}+c)\equiv e^{F}(H(x)+c_{2})} 8111: 7938:

Introducing every concept with as few technicalities as possible (in particular starting with explicit equations)

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as a practical illustration in his book. And so, I have linked the math to a Knowledge article on the subject.

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Berberan-Santos, Mário N. (2010). "Green's function method and the first-order linear differential equation".

3030: 245: 8263: 8253: 8158:"characteristic equation" is used repeatedly for differential equations on pp. 138, 139, 141, 145, 148, etc. 7276: 6737: 223:

The lead refers to both ODEs and PDEs. The solution is not to rename but to improve the article in my view.

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What do you prefer? Is there a clear majority among the editors of this article for one option or another?

7279:. If the merging problem is constantly discussed, I think that both of above cites should be merged into a 7051: 8068: 8035: 7512: 7427: 7141: 7115: 3599: 2164: 225: 8904:

In that case, we currently have at least 4 independent articles containing a walk-through of this method:

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The first method of solving linear ordinary differential equations with constant coefficients is due to

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The first method of solving linear ordinary differential equations with constant coefficients is due to

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still has to be differentiable!) But for those who know more, would that be a correct change to make?

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Specifically, for consistency with the introductory text it is more appropriate to use the exponential

2923: 1101: 94: 7481:- I claim that at this here level the term "value", when addressing a full blown "map", is disturbing. 7132:. Basically, there's less here than meets the eye. The current state of the target article contains 6523: 5975: 3219: 8164:"characteristic polynomial" is used repeatedly for matrices on pp. 334, 338, 339, 341, 342, 345, etc. 7526: 7043: 6978: 6717: 5360: 5234: 4287: 4244: 4230: 4161: 4147: 3320: 3024: 1077: 362: 341: 184: 8907: 8861: 8857: 8834: 7304: 7081: 21: 7823: 7804: 7085: 1096: 2063: 1554: 1205: 1152: 209: 116:

on Knowledge. If you would like to participate, please visit the project page, where you can join

8931: 8919: 8887: 8876: 8842: 8797: 8297: 8051: 8000: 7977: 7957: 7916: 7904: 7865:- Maybe the b(x) should be moved to the LHS of the equation for this polynomial view on the LDEQ. 7831: 7494:

converge to the desired level, leaving aside the already identified flaws of the current version.

7469:- The first sentence of the current lead is contained in my version, amended by a well deserved . 7413: 7393: 7241:

Next, from the mission and purpose of the Knowledge, I think our articles should be posted on it.

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arising in solving special cases of certain classes of DEQ. I also agree to the arguments above.

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I find this use to have strange look in the context. In particular how shall we write z? Is it

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depends on x and so it is not constant. Perhaps it could say "if the function b(x) is zero".

4017: 3794: 3476: 935: 593:{\displaystyle a_{n}(x)D^{n}y(x)+a_{n-1}(x)D^{n-1}y(x)+\cdots +a_{1}(x)Dy(x)+a_{0}(x)y(x)=f(x)} 8064: 8031: 7992: 7424: 7138: 7112: 6752: 5419: 4269: 2784: 257: 213: 8890:. And this article should mention the term "integrating factor" with a link to its article. 8231:"characteristic equation" is used repeatedly for matrices on pp. 320-327, 383, 385, 386, etc. 8088: 4847: 4506: 1277: 357:

Now I will look for some more interesting simple examples... such as electric circuits,... —

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for not simply honoring coining acronyms as justification for creating new WP articles on

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Elementary Differential Equations and Boundary Value Problems, 10th ed., Wiley, 2012.

8047: 7996: 7973: 7965: 7912: 7827: 7409: 7389: 7176: 6935:{\displaystyle P_{0}(x)y^{(n)}+P_{1}(x)y^{(n-1)}+\cdots +P_{n-1}(x)y'+P_{n}(x)y=F(x)} 5434: 1146: 8134:"characteristic equation" is used repeatedly for matrices on pp. 384, 385, 408, etc. 7931:

I have rewritten the whole article in the spirit of the new lead. The objective was

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Currently the section "First-order equation with variable coefficients" concludes

8207:"characteristic polynomial" is used for differential equations on pp. 88, 91, 126. 630:

are given functions. and the source term is considered to be a function of time ƒ(

6727:

Linear differential equations include inhomogeneous linear differential equations

8774: 8080: 7808: 7450: 7330: 6747:

Agreed. In support here are two quotations from C. H. Edwards and D. E. Penny,

5387: 113: 8506:

is already contained in the integral. If we say that H is an antiderivative of

6968:

Should the lower integration limit perhaps be zero rather than a? Help please.

5378:

Linear_differential_equation#Nonhomogeneous_equation_with_constant_coefficients

1938:{\displaystyle x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots +a_{0}y(x)=0.} 8175: 7023: 6963:

First Order Equations with Varying Coefficients, Example, alternative equation

380: 90: 8920:

Integrating factor#Solving first order linear ordinary differential equations

8888:

Integrating factor#Solving first order linear ordinary differential equations

7268:

How can you give a vast theory of "linear differential equation" to one page?

8916:

Linear differential equation#First-order equation with variable coefficients

8891: 8092: 7991:

IMHO, with the new version of the article, the rating should be upgraded to

7908: 7842: 6948: 8189:

neither "characteristic equation" nor "characteristic polynomial" are used.

7230:"Can not you use the ERF method if P (r) is 0? or what is it if we can use? 6411:{\displaystyle y_{1'}={A_{0}e^{ikx}-A_{1}e^{-ikx} \over 2i}=C_{1}\sin(kx).} 2768:{\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left{\frac {dw}{dz}}-abw=0.} 336:

is the homogeneous form. I will develop this theme on paper for a bit... —

7961: 7869:- The notion of the "order" of a DEQ should be mentioned within the lead. 6702:{\displaystyle y_{0'}={C_{0}e^{ikx}+C_{0}e^{-ikx} \over 2}=C_{0}\cos(kx)} 6264:{\displaystyle y_{0'}={A_{0}e^{ikx}+A_{1}e^{-ikx} \over 2}=C_{0}\cos(kx)} 7291:

Therefore, I think our article should exist in Knowledge independently.

2047:{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+ax{\frac {dy}{dx}}+by=0\,} 4996:{\displaystyle r^{N}e^{rt}+A_{1}r^{N-1}e^{rt}+\cdots +A_{N}e^{rt}=0.} 4655:{\displaystyle z^{n}e^{zx}+A_{1}z^{n-1}e^{zx}+\cdots +A_{n}e^{zx}=0.} 203:

Article Should be Re-titled: "Linear Ordinary Differential Equations"

8161:"characteristic equation" is used for matrices only once on pp. 351. 7498:

To make a discussion reasonably possible, I copied my version below.

659:

Now that I have tried to edit the introduction in terms of variable

7893:

For 2d and 4th points, these are tweaks that are easy to implement.

6764:... the general nth-order linear differential equation of the form 1404:{\displaystyle \ x^{2}y''+xy'+(x^{2}-n^{2})y=0,\ n\in \mathbb {R} } 5255: 5142:

can not be zero then we have the classic characteristic equation:

4306: 8837:

instead since there is clearly no material there worth merging.

8210:"characteristic polynomial" is used for matrices on pp. 160, 388. 5403:"In mathematics, a linear differential equation is of the form:" 7442: 7423:

It could be a good article, but on Wikibooks or Wikiversity. —

919:{\displaystyle g=p_{0}D^{2}f(x)+p_{1}D^{1}f(x)+p_{2}D^{0}f(x)\!} 8788:

In this context, an indefinite integral must be interpreted as

8773:

And then the constant simply gets absorbed into the integral.__

8245:

What do we want to do to clarify the terminology for students?

5400:

I hope no one minds that I'm changing the first sentence from:

391:. Therefore a fairly general form of such an equation would be 371:

I had commented out the following text in the original article

8182:

Vol. II/2, Chapter 6 "Differential Equations", Springer, 2000.

6947:

part of the article that deals with nonhomogeneous equations.

15: 3014:{\displaystyle x\,y''+(1-x)\,y'+ny=0,\ n\in \mathbb {N} _{0}} 8926:

my original merge proposal since there is nothing to merge.

8389:{\displaystyle {\frac {d}{dx}}\left(ye^{-F}\right)=ge^{-F}.} 648:

that is involved. (Assuming a possibly existing coefficient

387:; but permits, for example, taking the second derivative of 8198:

Differential Equations: From Calculus to Dynamical Systems,

7778:

which allow for closed solvability in the homogeneous case.

7365:, but -at my discretion- stop commenting on these matters. 5929:

I don't think the cosine form should have a denominator of

5102:{\displaystyle (r^{N}+A_{1}r^{N-1}+\cdots +A_{N})e^{rt}=0.} 4282:

Then I will add in other reference works as appropriate. —

8864:

just doubles what is already on this page, in the section

8087:

is that it can be applied to discontinuous functions (see

5621:

In particular, the following solutions can be constructed

1777:{\displaystyle \sum _{i=0}^{n}b_{i}(cx+d)^{i}y^{(i)}(x)=0} 8083:

function. It is said that an advantage of the method of

7105:

Linear differential equation#Exponential response formula

7903:

By the way, this article must also have a section about

6131:

These solutions provide a basis for the two-dimensional

5988:

Well spotted it was introduced in this edit in February

5448:

I agree with Sandrobt - it should be fixed. Example is:

7589:

of parameters that satisfies the differential equation.

7349:

There is a lot of heavy pseudo-activity in editing the

5989: 5374: 5753: 5650: 379:

rules out operations such as taking the square of the

8856:

There has been a new article created recently on the

8548: 8512: 8408: 8322: 7724: 7697: 7661: 7639: 7619: 7599: 7575: 7529: 7103:

be merged as at most one paragraph of explanation to

6770: 6582: 6526: 6430: 6283: 6144: 6070: 6004: 5958: 5935: 5900: 5868: 5836: 5630: 5558: 5528: 5488: 5454: 5319: 5294: 5264: 5151: 5118: 5015: 4889: 4850: 4709: 4672: 4548: 4509: 4368: 4345: 4315: 4204: 4177: 4020: 4001:{\displaystyle (1-x^{2})\,y''-3x\,y'+n(n+2)\,y=0\,\!} 3910: 3797: 3698: 3607: 3479: 3329: 3227: 3116: 3033: 2931: 2787: 2631: 2483: 2418: 2328: 2256: 2172: 2066: 1953: 1796: 1686: 1557: 1423: 1309: 1280: 1208: 1155: 1104: 938: 805: 788:

to explain the linear transformation of the function

711: 673: 400: 304: 260: 8822:

Subsequent comments should be made in a new section.

2237:{\displaystyle \ y''-2xy'+2ny=0,\ n\in \mathbb {Z} } 698:

differential forms I am beginning to think that the

112:, a collaborative effort to improve the coverage of 8085:

Laplace transform applied to differential equations

5219:{\displaystyle r^{N}+A_{1}r^{N-1}+\cdots +A_{N}=0.} 3778:{\displaystyle (1-x^{2})\,y''-x\,y'+n^{2}\,y=0\,\!} 3460:{\displaystyle {d \over dx}\left+n(n+1)P_{n}(x)=0.} 298:is a good starting position for a line in 2D. And, 169:

Moved material from ordinary differential equations

8762: 8531: 8471: 8388: 8106:Characteristic Equation, Characteristic Polynomial 7944:Explaineing how the different methods are related. 7739: 7709: 7679: 7645: 7625: 7605: 7581: 7561: 7489:- Raising to powers smaller than one is de rigeur? 6934: 6701: 6565: 6509:{\displaystyle y_{H}=C_{0}\cos(kx)+C_{1}\sin(kx).} 6508: 6410: 6263: 6118: 6046: 5964: 5944: 5918: 5886: 5854: 5814: 5603: 5544: 5514: 5474: 5370:Nonhomogeneous equation with constant coefficients 5340: 5300: 5280: 5218: 5134: 5101: 4995: 4872: 4833: 4688: 4654: 4531: 4492: 4351: 4331: 4217: 4190: 4128: 4000: 3894: 3777: 3678: 3585: 3459: 3309: 3205: 3096: 3013: 2909: 2767: 2611: 2461: 2402: 2308: 2236: 2150: 2046: 1937: 1776: 1664: 1537: 1403: 1286: 1266: 1188: 1135: 1057: 918: 777: 690: 592: 328: 290: 7950: 7354: 7285:However, there are exist two cites independently. 5409:"Linear differential equations are of the form:" 173:I moved a lot of material (mostly examples) from 8219:Introduction to Ordinary Differential Equations, 8089:Introduction to Laplace Transforms for Engineers 8079:example, in an electrical problem it could be a 7066:The first paragraph of the "Introduction" reads 6573:for equality to hold. It may be better to write 5616:Harmonic Oscillator equivalent solutions mangled 4297:Homogeneous equations with constant coefficients 8979:Knowledge level-5 vital articles in Mathematics 8866:First-order equation with variable coefficients 5993: 5380:(the edit is from when the section was part of 3996: 3773: 2398: 2146: 1660: 1262: 1184: 1053: 914: 773: 8149:Differential Equations and Their Applications, 7070:Linear differential equations are of the form 4268:, Cambridge, UK.: Cambridge University Press, 8825:A summary of the conclusions reached follows. 8472:{\displaystyle y=ce^{F}+e^{F}\int ge^{-F}dx,} 3310:{\displaystyle \ (1-x^{2})y''-2xy'+n(n+1)y=0} 8: 8269:We do nothing and let students in confusion. 7909:Dynamic Dictionary of Mathematical Functions 7807:and requested participation in this thread. 7220:ERF was mentioned in Knowledge cite, titled 5258:, who realized that solutions have the form 4309:, who realized that solutions have the form 7751:is of great importance. The solutions to ( 3679:{\displaystyle \ (1-x^{2})y''-xy'+n^{2}y=0} 1068:So! I think I will try to re-introduce the 778:{\displaystyle g=p_{0}f''+p_{1}f'+p_{2}f\!} 7041: 6976: 58: 8751: 8723: 8701: 8673: 8657: 8629: 8616: 8591: 8575: 8562: 8547: 8520: 8511: 8451: 8435: 8422: 8407: 8374: 8350: 8323: 8321: 8121:William E. Boyce and Richard C. DiPrima, 7723: 7696: 7660: 7638: 7618: 7598: 7574: 7547: 7537: 7528: 7254:Finally, let's discuss the merge problem. 6899: 6863: 6832: 6813: 6794: 6775: 6769: 6675: 6647: 6637: 6618: 6608: 6601: 6587: 6581: 6557: 6544: 6531: 6525: 6479: 6448: 6435: 6429: 6381: 6348: 6338: 6319: 6309: 6302: 6288: 6282: 6237: 6209: 6199: 6180: 6170: 6163: 6149: 6143: 6098: 6088: 6075: 6069: 6032: 6022: 6009: 6003: 5957: 5934: 5905: 5899: 5873: 5867: 5841: 5835: 5785: 5752: 5736: 5709: 5699: 5680: 5670: 5649: 5635: 5629: 5569: 5557: 5536: 5527: 5506: 5493: 5487: 5453: 5318: 5293: 5269: 5263: 5204: 5179: 5169: 5156: 5150: 5123: 5117: 5084: 5071: 5046: 5036: 5023: 5014: 4978: 4968: 4946: 4930: 4920: 4904: 4894: 4888: 4861: 4849: 4816: 4788: 4764: 4757: 4751: 4735: 4717: 4710: 4708: 4677: 4671: 4637: 4627: 4605: 4589: 4579: 4563: 4553: 4547: 4520: 4508: 4475: 4447: 4423: 4416: 4410: 4394: 4376: 4369: 4367: 4344: 4320: 4314: 4225:to make sure that no errors were made. — 4209: 4203: 4182: 4176: 4104: 4073: 4054: 4041: 4019: 3924: 3909: 3870: 3860: 3847: 3831: 3818: 3796: 3754: 3712: 3697: 3661: 3623: 3606: 3561: 3530: 3511: 3498: 3478: 3436: 3391: 3372: 3363: 3330: 3328: 3243: 3226: 3179: 3161: 3131: 3115: 3097:{\displaystyle x\,y''+(1-x)\,y'+n\,y=0\,} 3032: 3005: 3001: 3000: 2930: 2887: 2868: 2815: 2786: 2727: 2675: 2657: 2650: 2630: 2605: 2604: 2482: 2417: 2376: 2357: 2338: 2327: 2255: 2230: 2229: 2171: 2121: 2105: 2086: 2076: 2065: 2007: 1989: 1971: 1964: 1958: 1952: 1911: 1871: 1855: 1839: 1811: 1801: 1795: 1747: 1737: 1712: 1702: 1691: 1685: 1635: 1622: 1609: 1593: 1577: 1567: 1556: 1517: 1504: 1474: 1459: 1441: 1434: 1428: 1422: 1397: 1396: 1367: 1354: 1316: 1308: 1279: 1237: 1218: 1207: 1154: 1103: 1008: 998: 985: 975: 962: 952: 937: 929:and then with a little rewriting we have 896: 886: 861: 851: 826: 816: 804: 764: 743: 722: 710: 677: 672: 644:, the index of the highest derivative of 548: 511: 474: 449: 424: 405: 399: 303: 259: 7324:, mainly due to the face that I believe 5991:. I've reverted it. The current text is 5229:So! This is what I propose to do next. — 8969:Knowledge vital articles in Mathematics 7755:) linear differential equations form a 7653:, the LDEQ can compactly be written as 7002: 4115: 4026: 3994: 3983: 3952: 3933: 3881: 3803: 3771: 3760: 3737: 3721: 3572: 3201: 3190: 3173: 3155: 3125: 3092: 3081: 3065: 3037: 2963: 2935: 2476:Hypergeometrische Differentialgleichung 2387: 2144: 2042: 1658: 1260: 1182: 60: 19: 8180:Introduction to Calculus and Analysis, 7766:Linear differential equations can be 7062:"Of the Same Nature" phrase is unclear 6520:which is still not perfect as we need 5552:is not solution of original equation: 3600:Tschebyschowsche Differentialgleichung 8984:C-Class vital articles in Mathematics 4301:Now I want to tidy up the following: 7: 8816:The following discussion is closed. 6119:{\displaystyle y_{1}=A_{1}e^{-ikx}.} 2623:Hypergeometric differential equation 2462:{\displaystyle u''-2xu'=-2\lambda u} 2322:(from the equation on German site): 2309:{\displaystyle L=u''-xu'=-\lambda u} 106:This article is within the scope of 7956:Finally, I have added a section on 7273:Method of undetermined coefficients 7107:. See also previous discussion at 5604:{\displaystyle (2x^{2})'=4x\neq 2x} 4266:The Nature of Mathematical Modeling 181:18:21, 18 March 2007 (UTC) Thanks 49:It is of interest to the following 8994:High-priority mathematics articles 8833:by the nominator. I have PROD'ded 8486:is a constant of integration, and 8200:2nd ed., Vol. 43, MAA Press, 2019. 6047:{\displaystyle y_{0}=A_{0}e^{ikx}} 3220:Legendresche Differentialgleichung 2924:Laguerresche Differentialgleichung 1136:{\displaystyle \ y''-\lambda xy=0} 14: 7761:non-linear differential equations 7109:Talk:Exponential response formula 7035:Introductory Paragraph Misleading 6749:Elementary Differential Equations 6566:{\displaystyle A_{0}=A_{1}=C_{0}} 5995:The solutions are, respectively, 5288:, for possibly-complex values of 4339:, for possibly-complex values of 2165:Hermitesche Differentialgleichung 637:Such an equation is said to have 126:Knowledge:WikiProject Mathematics 8964:Knowledge level-5 vital articles 8941:The discussion above is closed. 7562:{\displaystyle (y_{i})_{i\in I}} 1302:Besselsche Differentialgleichung 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 8308:Redundant integration constant? 7361:I'll put this note also in the 1679:Eulersche Differentialgleichung 175:ordinary differential equations 146:This article has been rated as 8974:C-Class level-5 vital articles 8757: 8741: 8735: 8729: 8713: 8691: 8685: 8679: 8663: 8647: 8641: 8635: 8399:Thus, the general solution is 7544: 7530: 6991:13:55, 16 September 2022 (UTC) 6929: 6923: 6911: 6905: 6881: 6875: 6845: 6833: 6825: 6819: 6801: 6795: 6787: 6781: 6696: 6687: 6500: 6491: 6469: 6460: 6402: 6393: 6258: 6249: 5806: 5797: 5576: 5559: 5382:Ordinary_differential_equation 5365:08:25, 15 September 2009 (UTC) 5239:14:33, 14 September 2009 (UTC) 5077: 5016: 4292:13:12, 14 September 2009 (UTC) 4249:20:14, 16 September 2009 (UTC) 4235:20:14, 16 September 2009 (UTC) 4166:18:50, 16 September 2009 (UTC) 4152:17:41, 16 September 2009 (UTC) 4097: 4085: 4047: 4028: 3980: 3968: 3930: 3911: 3824: 3805: 3718: 3699: 3629: 3610: 3554: 3542: 3504: 3485: 3448: 3442: 3429: 3417: 3403: 3397: 3369: 3350: 3295: 3283: 3249: 3230: 3152: 3140: 3062: 3050: 2960: 2948: 2853: 2835: 2808: 2796: 2716: 2698: 2647: 2635: 2538: 2520: 2501: 2489: 2266: 2260: 1926: 1920: 1895: 1889: 1884: 1872: 1829: 1823: 1818: 1812: 1765: 1759: 1754: 1748: 1734: 1718: 1628: 1602: 1523: 1497: 1373: 1347: 1097:Airysche Differentialgleichung 1082:17:05, 16 September 2009 (UTC) 1050: 1047: 1041: 1035: 1026: 1020: 1014: 945: 911: 905: 876: 870: 841: 835: 587: 581: 572: 566: 560: 554: 538: 532: 523: 517: 495: 489: 467: 461: 439: 433: 417: 411: 367:11:49, 15 September 2009 (UTC) 346:09:36, 15 September 2009 (UTC) 250:15:29, 26 September 2008 (UTC) 1: 8802:11:43, 12 February 2021 (UTC) 8783:11:01, 12 February 2021 (UTC) 8101:10:20, 1 September 2018 (UTC) 8005:07:34, 28 February 2018 (UTC) 7509:Linear differential equations 7056:10:34, 19 November 2015 (UTC) 6742:16:04, 30 December 2013 (UTC) 5391:07:16, 18 November 2010 (UTC) 2151:{\displaystyle \lefty=0,\,\!} 1665:{\displaystyle \lefty=0,\,\!} 1274:(notice the inclusion of the 1267:{\displaystyle \lefty=0,\,\!} 1189:{\displaystyle y''-xy=0,\,\!} 608:is the differential operator 197:02:38, 16 February 2008 (UTC) 120:and see a list of open tasks. 8989:C-Class mathematics articles 8936:20:26, 13 January 2022 (UTC) 8847:21:30, 13 January 2022 (UTC) 7982:17:44, 6 February 2018 (UTC) 7921:12:05, 28 January 2018 (UTC) 7885:10:56, 28 January 2018 (UTC) 7851:21:03, 27 January 2018 (UTC) 7836:18:14, 27 January 2018 (UTC) 7817:17:20, 27 January 2018 (UTC) 7798:09:51, 27 January 2018 (UTC) 7613:, the unknown function with 7459:21:00, 26 January 2018 (UTC) 7345:Obstruction to pending merge 7320:, per my explanation on the 7281:linear differential equation 7261:Linear differential equation 7222:Linear differential equation 7101:Exponential response formula 7095:Exponential response formula 6751:, 4ed, Prentice-Hall, 2000, 5244:Doctors differ, patients die 3206:{\displaystyle \left\,y=0\,} 2403:{\displaystyle \left\,y=0\!} 8900:21:32, 6 January 2022 (UTC) 8881:18:55, 6 January 2022 (UTC) 8073:23:35, 27 August 2018 (UTC) 7211:Mr. Rubin and Mr. Prugy, I 5515:{\displaystyle y_{0}=x^{2}} 5482:, one of it's solutions is 5443:05:11, 9 January 2013 (UTC) 375:The linearity condition on 9010: 6957:16:48, 24 March 2015 (UTC) 4264:Gershenfeld, Neil (1999), 4129:{\displaystyle \left\,y=0} 3895:{\displaystyle \left\,y=0} 3586:{\displaystyle \left\,y=0} 1058:{\displaystyle g=f(x)=L\!} 8302:12:03, 13 July 2020 (UTC) 8283:11:25, 13 July 2020 (UTC) 8112:Characteristic polynomial 8056:20:40, 20 July 2018 (UTC) 8040:19:54, 20 July 2018 (UTC) 7363:other article's talk page 7339:07:55, 24 June 2017 (UTC) 7311:) 2:38, 6 June 2017 (UTC) 7024:10.1007/s10910-010-9678-2 6732:article to reflect this. 2910:{\displaystyle \leftw=0.} 796:. So let us compare with 291:{\displaystyle ax+by+c=0} 230:21:17, 15 June 2008 (UTC) 218:19:19, 13 June 2008 (UTC) 145: 78: 57: 8943:Please do not modify it. 8886:A merge could made with 8819:Please do not modify it. 8151:4th ed., Springer, 1993. 8010:Continuous coefficients? 7859:Thoughts on the new lead 7431:21:13, 8 June 2017 (UTC) 7418:13:39, 3 June 2017 (UTC) 7398:13:39, 3 June 2017 (UTC) 7375:07:49, 3 June 2017 (UTC) 7203:06:39, 31 May 2017 (UTC) 7185:19:15, 30 May 2017 (UTC) 7167:06:56, 30 May 2017 (UTC) 7145:05:39, 30 May 2017 (UTC) 7119:05:39, 30 May 2017 (UTC) 6722:04:28, 22 May 2013 (UTC) 5984:02:37, 22 May 2013 (UTC) 5424:17:09, 11 May 2011 (UTC) 4873:{\displaystyle y=e^{rt}} 4532:{\displaystyle y=e^{zx}} 1287:{\displaystyle \lambda } 152:project's priority scale 8532:{\displaystyle ge^{-F}} 8264:Characteristic equation 8254:Characteristic equation 7277:Variation of parameters 7090:17:35, 1 May 2016 (UTC) 5919:{\displaystyle y_{0''}} 5348:? Not is this context! 4696:as a function of time. 329:{\displaystyle ax+by=0} 109:WikiProject Mathematics 8959:C-Class vital articles 8764: 8533: 8473: 8390: 7927:Rewrite of the article 7741: 7711: 7681: 7647: 7633:and the response with 7627: 7607: 7583: 7563: 7513:differential equations 6936: 6703: 6567: 6518: 6510: 6412: 6265: 6120: 6048: 5966: 5946: 5920: 5888: 5887:{\displaystyle y_{1'}} 5856: 5855:{\displaystyle y_{0'}} 5824: 5816: 5605: 5546: 5545:{\displaystyle 2y_{0}} 5516: 5476: 5342: 5341:{\displaystyle z=x+iy} 5311: 5302: 5282: 5281:{\displaystyle e^{zx}} 5220: 5136: 5135:{\displaystyle e^{rt}} 5103: 4997: 4874: 4835: 4690: 4689:{\displaystyle e^{rt}} 4664: 4656: 4533: 4494: 4353: 4333: 4332:{\displaystyle e^{zx}} 4261: 4219: 4192: 4130: 4002: 3896: 3779: 3680: 3587: 3461: 3311: 3207: 3098: 3015: 2911: 2769: 2613: 2463: 2404: 2310: 2238: 2152: 2048: 1939: 1778: 1707: 1666: 1539: 1405: 1288: 1268: 1190: 1137: 1059: 920: 779: 692: 667:and used conventional 657: 594: 330: 292: 8765: 8534: 8474: 8391: 8221:4th ed., Wiley, 1989. 7987:Rating of the article 7776:constant coefficients 7742: 7712: 7682: 7680:{\displaystyle Ly=f.} 7648: 7628: 7608: 7584: 7564: 7437:Major rewrite of lead 6937: 6704: 6568: 6511: 6413: 6266: 6121: 6049: 5967: 5965:{\displaystyle \cos } 5947: 5921: 5889: 5857: 5817: 5619: 5606: 5547: 5517: 5477: 5475:{\displaystyle y'=2x} 5343: 5303: 5283: 5251: 5221: 5137: 5104: 4998: 4875: 4836: 4691: 4657: 4534: 4495: 4354: 4334: 4303: 4220: 4218:{\displaystyle D^{0}} 4193: 4191:{\displaystyle D^{1}} 4131: 4003: 3897: 3780: 3690:Chebyshev polynomials 3681: 3588: 3462: 3312: 3208: 3099: 3016: 2912: 2770: 2614: 2464: 2405: 2311: 2248:Hermitian polynomials 2239: 2153: 2049: 1940: 1788:Euler-Cauchy equation 1779: 1687: 1667: 1540: 1406: 1289: 1269: 1191: 1138: 1060: 921: 780: 693: 595: 373: 331: 293: 36:level-5 vital article 8546: 8510: 8502:It seems to me that 8406: 8320: 8248:I see four options: 7740:{\displaystyle Ly=0} 7722: 7695: 7659: 7637: 7617: 7597: 7573: 7527: 7351:article to be merged 7322:talk page discussion 6768: 6580: 6524: 6428: 6281: 6142: 6068: 6002: 5956: 5933: 5898: 5866: 5834: 5628: 5556: 5526: 5486: 5452: 5317: 5292: 5262: 5149: 5116: 5013: 5006:and this factors as 4887: 4848: 4707: 4670: 4546: 4507: 4366: 4343: 4313: 4202: 4175: 4156:Taking a break :) — 4018: 4012:Linear operator form 3908: 3795: 3789:Linear operator form 3696: 3605: 3477: 3471:Linear operator form 3327: 3321:Legendre polynomials 3225: 3114: 3108:Linear operator form 3031: 3025:Laguerre polynomials 2929: 2785: 2781:(English version) : 2779:Linear operator form 2629: 2481: 2416: 2326: 2320:Linear operator form 2254: 2170: 2064: 2058:Linear operator form 1951: 1794: 1684: 1555: 1549:Linear operator form 1421: 1307: 1278: 1206: 1200:Linear operator form 1153: 1102: 936: 803: 709: 691:{\displaystyle d/dt} 671: 398: 302: 258: 132:mathematics articles 8196:Virginia Noonburg, 7958:holonomic functions 7905:holonomic functions 7710:{\displaystyle f=0} 5952:in the argument of 1947:and also the form : 8760: 8529: 8469: 8386: 8275:J.P. Martin-Flatin 7737: 7707: 7677: 7643: 7623: 7603: 7579: 7559: 6932: 6699: 6563: 6506: 6408: 6261: 6116: 6044: 5962: 5945:{\displaystyle 2i} 5942: 5916: 5884: 5852: 5812: 5772: 5659: 5601: 5542: 5512: 5472: 5338: 5298: 5278: 5216: 5132: 5099: 4993: 4870: 4831: 4686: 4652: 4529: 4490: 4349: 4329: 4215: 4188: 4126: 4116: 4027: 3998: 3997: 3995: 3984: 3953: 3934: 3892: 3882: 3804: 3775: 3774: 3772: 3761: 3738: 3722: 3676: 3583: 3573: 3457: 3307: 3203: 3202: 3191: 3174: 3156: 3126: 3094: 3093: 3082: 3066: 3038: 3011: 2964: 2936: 2907: 2765: 2609: 2459: 2400: 2399: 2388: 2306: 2234: 2148: 2147: 2145: 2044: 2043: 1935: 1774: 1662: 1661: 1659: 1535: 1401: 1284: 1264: 1263: 1261: 1186: 1185: 1183: 1133: 1087:Classical examples 1055: 1054: 916: 915: 775: 774: 688: 590: 326: 288: 101:Mathematics portal 45:content assessment 8336: 8217:Shepley L. Ross, 7788:End of my reply. 7646:{\displaystyle f} 7626:{\displaystyle y} 7606:{\displaystyle L} 7582:{\displaystyle I} 7235:on the Knowledge. 7058: 7046:comment added by 6993: 6981:comment added by 6666: 6372: 6228: 5771: 5658: 5301:{\displaystyle z} 4801: 4742: 4460: 4401: 4352:{\displaystyle z} 3609: 3385: 3343: 3229: 2992: 2745: 2682: 2585: 2485: 2222: 2174: 2025: 1996: 1492: 1466: 1389: 1311: 1106: 199: 187:comment added by 166: 165: 162: 161: 158: 157: 9001: 8821: 8769: 8767: 8766: 8761: 8756: 8755: 8728: 8727: 8706: 8705: 8678: 8677: 8662: 8661: 8634: 8633: 8621: 8620: 8599: 8598: 8580: 8579: 8567: 8566: 8538: 8536: 8535: 8530: 8528: 8527: 8498: 8485: 8478: 8476: 8475: 8470: 8459: 8458: 8440: 8439: 8427: 8426: 8395: 8393: 8392: 8387: 8382: 8381: 8363: 8359: 8358: 8357: 8337: 8335: 8324: 8291: 7951:above discussion 7746: 7744: 7743: 7738: 7716: 7714: 7713: 7708: 7686: 7684: 7683: 7678: 7652: 7650: 7649: 7644: 7632: 7630: 7629: 7624: 7612: 7610: 7609: 7604: 7588: 7586: 7585: 7580: 7568: 7566: 7565: 7560: 7558: 7557: 7542: 7541: 7443:Purgy Purgatorio 7387: 7384:Purgy Purgatorio 7028: 7027: 7007: 6941: 6939: 6938: 6933: 6904: 6903: 6891: 6874: 6873: 6849: 6848: 6818: 6817: 6805: 6804: 6780: 6779: 6708: 6706: 6705: 6700: 6680: 6679: 6667: 6662: 6661: 6660: 6642: 6641: 6629: 6628: 6613: 6612: 6602: 6597: 6596: 6595: 6572: 6570: 6569: 6564: 6562: 6561: 6549: 6548: 6536: 6535: 6515: 6513: 6512: 6507: 6484: 6483: 6453: 6452: 6440: 6439: 6417: 6415: 6414: 6409: 6386: 6385: 6373: 6371: 6363: 6362: 6361: 6343: 6342: 6330: 6329: 6314: 6313: 6303: 6298: 6297: 6296: 6270: 6268: 6267: 6262: 6242: 6241: 6229: 6224: 6223: 6222: 6204: 6203: 6191: 6190: 6175: 6174: 6164: 6159: 6158: 6157: 6125: 6123: 6122: 6117: 6112: 6111: 6093: 6092: 6080: 6079: 6053: 6051: 6050: 6045: 6043: 6042: 6027: 6026: 6014: 6013: 5971: 5969: 5968: 5963: 5951: 5949: 5948: 5943: 5925: 5923: 5922: 5917: 5915: 5914: 5913: 5893: 5891: 5890: 5885: 5883: 5882: 5881: 5861: 5859: 5858: 5853: 5851: 5850: 5849: 5821: 5819: 5818: 5813: 5790: 5789: 5777: 5773: 5770: 5762: 5754: 5741: 5740: 5728: 5724: 5723: 5722: 5704: 5703: 5691: 5690: 5675: 5674: 5660: 5651: 5645: 5644: 5643: 5610: 5608: 5607: 5602: 5582: 5574: 5573: 5551: 5549: 5548: 5543: 5541: 5540: 5521: 5519: 5518: 5513: 5511: 5510: 5498: 5497: 5481: 5479: 5478: 5473: 5462: 5347: 5345: 5344: 5339: 5307: 5305: 5304: 5299: 5287: 5285: 5284: 5279: 5277: 5276: 5225: 5223: 5222: 5217: 5209: 5208: 5190: 5189: 5174: 5173: 5161: 5160: 5141: 5139: 5138: 5133: 5131: 5130: 5108: 5106: 5105: 5100: 5092: 5091: 5076: 5075: 5057: 5056: 5041: 5040: 5028: 5027: 5002: 5000: 4999: 4994: 4986: 4985: 4973: 4972: 4954: 4953: 4941: 4940: 4925: 4924: 4912: 4911: 4899: 4898: 4879: 4877: 4876: 4871: 4869: 4868: 4840: 4838: 4837: 4832: 4821: 4820: 4802: 4800: 4799: 4798: 4779: 4775: 4774: 4758: 4756: 4755: 4743: 4741: 4740: 4739: 4726: 4722: 4721: 4711: 4695: 4693: 4692: 4687: 4685: 4684: 4661: 4659: 4658: 4653: 4645: 4644: 4632: 4631: 4613: 4612: 4600: 4599: 4584: 4583: 4571: 4570: 4558: 4557: 4538: 4536: 4535: 4530: 4528: 4527: 4499: 4497: 4496: 4491: 4480: 4479: 4461: 4459: 4458: 4457: 4438: 4434: 4433: 4417: 4415: 4414: 4402: 4400: 4399: 4398: 4385: 4381: 4380: 4370: 4358: 4356: 4355: 4350: 4338: 4336: 4335: 4330: 4328: 4327: 4277: 4224: 4222: 4221: 4216: 4214: 4213: 4197: 4195: 4194: 4189: 4187: 4186: 4135: 4133: 4132: 4127: 4114: 4110: 4109: 4108: 4078: 4077: 4059: 4058: 4046: 4045: 4007: 4005: 4004: 3999: 3961: 3942: 3929: 3928: 3901: 3899: 3898: 3893: 3880: 3876: 3875: 3874: 3865: 3864: 3852: 3851: 3836: 3835: 3823: 3822: 3784: 3782: 3781: 3776: 3759: 3758: 3746: 3730: 3717: 3716: 3685: 3683: 3682: 3677: 3666: 3665: 3653: 3639: 3628: 3627: 3608: 3592: 3590: 3589: 3584: 3571: 3567: 3566: 3565: 3535: 3534: 3516: 3515: 3503: 3502: 3473:(German site) : 3466: 3464: 3463: 3458: 3441: 3440: 3410: 3406: 3396: 3395: 3386: 3384: 3373: 3368: 3367: 3344: 3342: 3331: 3316: 3314: 3313: 3308: 3276: 3259: 3248: 3247: 3228: 3212: 3210: 3209: 3204: 3189: 3185: 3184: 3183: 3166: 3165: 3136: 3135: 3103: 3101: 3100: 3095: 3074: 3046: 3020: 3018: 3017: 3012: 3010: 3009: 3004: 2991: 2972: 2944: 2916: 2914: 2913: 2908: 2897: 2893: 2892: 2891: 2873: 2872: 2863: 2859: 2820: 2819: 2774: 2772: 2771: 2766: 2746: 2744: 2736: 2728: 2726: 2722: 2683: 2681: 2680: 2679: 2666: 2662: 2661: 2651: 2618: 2616: 2615: 2610: 2608: 2584: 2562: 2554: 2550: 2511: 2484: 2468: 2466: 2465: 2460: 2443: 2426: 2409: 2407: 2406: 2401: 2386: 2382: 2381: 2380: 2362: 2361: 2343: 2342: 2315: 2313: 2312: 2307: 2293: 2279: 2243: 2241: 2240: 2235: 2233: 2221: 2199: 2182: 2173: 2157: 2155: 2154: 2149: 2131: 2127: 2126: 2125: 2110: 2109: 2091: 2090: 2081: 2080: 2053: 2051: 2050: 2045: 2026: 2024: 2016: 2008: 1997: 1995: 1994: 1993: 1980: 1976: 1975: 1965: 1963: 1962: 1944: 1942: 1941: 1936: 1916: 1915: 1888: 1887: 1866: 1865: 1850: 1849: 1822: 1821: 1806: 1805: 1783: 1781: 1780: 1775: 1758: 1757: 1742: 1741: 1717: 1716: 1706: 1701: 1671: 1669: 1668: 1663: 1645: 1641: 1640: 1639: 1627: 1626: 1614: 1613: 1598: 1597: 1582: 1581: 1572: 1571: 1544: 1542: 1541: 1536: 1522: 1521: 1509: 1508: 1493: 1491: 1483: 1475: 1467: 1465: 1464: 1463: 1450: 1446: 1445: 1435: 1433: 1432: 1415:Bessel function 1410: 1408: 1407: 1402: 1400: 1388: 1372: 1371: 1359: 1358: 1343: 1329: 1321: 1320: 1310: 1293: 1291: 1290: 1285: 1273: 1271: 1270: 1265: 1247: 1243: 1242: 1241: 1223: 1222: 1195: 1193: 1192: 1187: 1163: 1142: 1140: 1139: 1134: 1114: 1105: 1064: 1062: 1061: 1056: 1013: 1012: 1003: 1002: 990: 989: 980: 979: 967: 966: 957: 956: 925: 923: 922: 917: 901: 900: 891: 890: 866: 865: 856: 855: 831: 830: 821: 820: 784: 782: 781: 776: 769: 768: 756: 748: 747: 735: 727: 726: 697: 695: 694: 689: 681: 599: 597: 596: 591: 553: 552: 516: 515: 485: 484: 460: 459: 429: 428: 410: 409: 335: 333: 332: 327: 297: 295: 294: 289: 182: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 9009: 9008: 9004: 9003: 9002: 9000: 8999: 8998: 8949: 8948: 8947: 8946: 8854: 8817: 8810: 8747: 8719: 8697: 8669: 8653: 8625: 8612: 8587: 8571: 8558: 8544: 8543: 8516: 8508: 8507: 8487: 8483: 8447: 8431: 8418: 8404: 8403: 8370: 8346: 8342: 8338: 8328: 8318: 8317: 8310: 8289: 8172:Richard Courant 8108: 8012: 7989: 7929: 7861: 7720: 7719: 7693: 7692: 7657: 7656: 7635: 7634: 7615: 7614: 7595: 7594: 7571: 7570: 7543: 7533: 7525: 7524: 7439: 7406: 7381: 7347: 7126: 7099:I propose that 7097: 7076: 7064: 7037: 7032: 7031: 7009: 7008: 7004: 6999: 6965: 6895: 6884: 6859: 6828: 6809: 6790: 6771: 6766: 6765: 6729: 6671: 6643: 6633: 6614: 6604: 6603: 6588: 6583: 6578: 6577: 6553: 6540: 6527: 6522: 6521: 6475: 6444: 6431: 6426: 6425: 6377: 6364: 6344: 6334: 6315: 6305: 6304: 6289: 6284: 6279: 6278: 6233: 6205: 6195: 6176: 6166: 6165: 6150: 6145: 6140: 6139: 6094: 6084: 6071: 6066: 6065: 6028: 6018: 6005: 6000: 5999: 5954: 5953: 5931: 5930: 5906: 5901: 5896: 5895: 5874: 5869: 5864: 5863: 5842: 5837: 5832: 5831: 5781: 5763: 5755: 5748: 5732: 5705: 5695: 5676: 5666: 5665: 5661: 5636: 5631: 5626: 5625: 5618: 5575: 5565: 5554: 5553: 5532: 5524: 5523: 5502: 5489: 5484: 5483: 5455: 5450: 5449: 5431: 5398: 5396:Very minor edit 5372: 5315: 5314: 5290: 5289: 5265: 5260: 5259: 5246: 5200: 5175: 5165: 5152: 5147: 5146: 5119: 5114: 5113: 5080: 5067: 5042: 5032: 5019: 5011: 5010: 4974: 4964: 4942: 4926: 4916: 4900: 4890: 4885: 4884: 4857: 4846: 4845: 4812: 4784: 4780: 4760: 4759: 4747: 4731: 4727: 4713: 4712: 4705: 4704: 4673: 4668: 4667: 4633: 4623: 4601: 4585: 4575: 4559: 4549: 4544: 4543: 4516: 4505: 4504: 4471: 4443: 4439: 4419: 4418: 4406: 4390: 4386: 4372: 4371: 4364: 4363: 4341: 4340: 4316: 4311: 4310: 4299: 4280: 4275: 4274:978-0521-570954 4263: 4256: 4205: 4200: 4199: 4178: 4173: 4172: 4100: 4069: 4050: 4037: 4025: 4021: 4016: 4015: 3954: 3935: 3920: 3906: 3905: 3866: 3856: 3843: 3827: 3814: 3802: 3798: 3793: 3792: 3750: 3739: 3723: 3708: 3694: 3693: 3657: 3646: 3632: 3619: 3603: 3602: 3557: 3526: 3507: 3494: 3484: 3480: 3475: 3474: 3432: 3387: 3377: 3359: 3349: 3345: 3335: 3325: 3324: 3269: 3252: 3239: 3223: 3222: 3175: 3157: 3127: 3121: 3117: 3112: 3111: 3067: 3039: 3029: 3028: 2999: 2965: 2937: 2927: 2926: 2883: 2864: 2828: 2824: 2811: 2792: 2788: 2783: 2782: 2737: 2729: 2691: 2687: 2671: 2667: 2653: 2652: 2627: 2626: 2555: 2519: 2515: 2504: 2479: 2478: 2436: 2419: 2414: 2413: 2372: 2353: 2334: 2333: 2329: 2324: 2323: 2286: 2272: 2252: 2251: 2192: 2175: 2168: 2167: 2117: 2101: 2082: 2072: 2071: 2067: 2062: 2061: 2017: 2009: 1985: 1981: 1967: 1966: 1954: 1949: 1948: 1907: 1867: 1851: 1835: 1807: 1797: 1792: 1791: 1743: 1733: 1708: 1682: 1681: 1631: 1618: 1605: 1589: 1573: 1563: 1562: 1558: 1553: 1552: 1513: 1500: 1484: 1476: 1455: 1451: 1437: 1436: 1424: 1419: 1418: 1363: 1350: 1336: 1322: 1312: 1305: 1304: 1276: 1275: 1233: 1214: 1213: 1209: 1204: 1203: 1156: 1151: 1150: 1107: 1100: 1099: 1089: 1004: 994: 981: 971: 958: 948: 934: 933: 892: 882: 857: 847: 822: 812: 801: 800: 760: 749: 739: 728: 718: 707: 706: 669: 668: 653: 628: 544: 507: 470: 445: 420: 401: 396: 395: 352:Shroud of Turin 300: 299: 256: 255: 238: 205: 171: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 9007: 9005: 8997: 8996: 8991: 8986: 8981: 8976: 8971: 8966: 8961: 8951: 8950: 8940: 8939: 8938: 8908:Armour formula 8905: 8902: 8862:Armour formula 8858:Armour formula 8853: 8852: 8851: 8850: 8849: 8835:Armour formula 8812: 8811: 8809: 8808:Proposed merge 8806: 8805: 8804: 8771: 8770: 8759: 8754: 8750: 8746: 8743: 8740: 8737: 8734: 8731: 8726: 8722: 8718: 8715: 8712: 8709: 8704: 8700: 8696: 8693: 8690: 8687: 8684: 8681: 8676: 8672: 8668: 8665: 8660: 8656: 8652: 8649: 8646: 8643: 8640: 8637: 8632: 8628: 8624: 8619: 8615: 8611: 8608: 8605: 8602: 8597: 8594: 8590: 8586: 8583: 8578: 8574: 8570: 8565: 8561: 8557: 8554: 8551: 8526: 8523: 8519: 8515: 8480: 8479: 8468: 8465: 8462: 8457: 8454: 8450: 8446: 8443: 8438: 8434: 8430: 8425: 8421: 8417: 8414: 8411: 8397: 8396: 8385: 8380: 8377: 8373: 8369: 8366: 8362: 8356: 8353: 8349: 8345: 8341: 8334: 8331: 8327: 8309: 8306: 8305: 8304: 8271: 8270: 8267: 8260: 8257: 8235: 8234: 8233: 8232: 8229: 8223: 8222: 8214: 8213: 8212: 8211: 8208: 8202: 8201: 8193: 8192: 8191: 8190: 8184: 8183: 8168: 8167: 8166: 8165: 8162: 8159: 8153: 8152: 8147:Martin Braun, 8144: 8143: 8142: 8141: 8138: 8135: 8132: 8126: 8125: 8107: 8104: 8076: 8075: 8059: 8058: 8011: 8008: 7988: 7985: 7946: 7945: 7942: 7939: 7936: 7928: 7925: 7924: 7923: 7901: 7897: 7894: 7891: 7870: 7868: 7866: 7864: 7860: 7857: 7856: 7855: 7854: 7853: 7801: 7800: 7785: 7784: 7783: 7782: 7781: 7780: 7779: 7764: 7749: 7748: 7747: 7736: 7733: 7730: 7727: 7706: 7703: 7700: 7689: 7688: 7687: 7676: 7673: 7670: 7667: 7664: 7642: 7622: 7602: 7590: 7578: 7556: 7553: 7550: 7546: 7540: 7536: 7532: 7501: 7500: 7499: 7496: 7495: 7491: 7490: 7487: 7486: 7483: 7482: 7479: 7478: 7475: 7474: 7471: 7470: 7467: 7466: 7441:I’ve reverted 7438: 7435: 7434: 7433: 7405: 7402: 7401: 7400: 7360: 7355:proposed merge 7346: 7343: 7342: 7341: 7313: 7312: 7302: 7298: 7297: 7293: 7292: 7288: 7287: 7269: 7265: 7264: 7257: 7256: 7250: 7249: 7244: 7243: 7237: 7236: 7232: 7226: 7225: 7217: 7216: 7208: 7207: 7206: 7205: 7188: 7187: 7169: 7147: 7125: 7122: 7096: 7093: 7068: 7063: 7060: 7036: 7033: 7030: 7029: 7018:(2): 175–178. 7001: 7000: 6998: 6995: 6983:128.253.104.13 6964: 6961: 6960: 6959: 6944: 6943: 6942: 6931: 6928: 6925: 6922: 6919: 6916: 6913: 6910: 6907: 6902: 6898: 6894: 6890: 6887: 6883: 6880: 6877: 6872: 6869: 6866: 6862: 6858: 6855: 6852: 6847: 6844: 6841: 6838: 6835: 6831: 6827: 6824: 6821: 6816: 6812: 6808: 6803: 6800: 6797: 6793: 6789: 6786: 6783: 6778: 6774: 6762: 6728: 6725: 6710: 6709: 6698: 6695: 6692: 6689: 6686: 6683: 6678: 6674: 6670: 6665: 6659: 6656: 6653: 6650: 6646: 6640: 6636: 6632: 6627: 6624: 6621: 6617: 6611: 6607: 6600: 6594: 6591: 6586: 6560: 6556: 6552: 6547: 6543: 6539: 6534: 6530: 6517: 6516: 6505: 6502: 6499: 6496: 6493: 6490: 6487: 6482: 6478: 6474: 6471: 6468: 6465: 6462: 6459: 6456: 6451: 6447: 6443: 6438: 6434: 6419: 6418: 6407: 6404: 6401: 6398: 6395: 6392: 6389: 6384: 6380: 6376: 6370: 6367: 6360: 6357: 6354: 6351: 6347: 6341: 6337: 6333: 6328: 6325: 6322: 6318: 6312: 6308: 6301: 6295: 6292: 6287: 6272: 6271: 6260: 6257: 6254: 6251: 6248: 6245: 6240: 6236: 6232: 6227: 6221: 6218: 6215: 6212: 6208: 6202: 6198: 6194: 6189: 6186: 6183: 6179: 6173: 6169: 6162: 6156: 6153: 6148: 6133:solution space 6129: 6128: 6127: 6126: 6115: 6110: 6107: 6104: 6101: 6097: 6091: 6087: 6083: 6078: 6074: 6060: 6059: 6055: 6054: 6041: 6038: 6035: 6031: 6025: 6021: 6017: 6012: 6008: 5961: 5941: 5938: 5912: 5909: 5904: 5880: 5877: 5872: 5848: 5845: 5840: 5823: 5822: 5811: 5808: 5805: 5802: 5799: 5796: 5793: 5788: 5784: 5780: 5776: 5769: 5766: 5761: 5758: 5751: 5747: 5744: 5739: 5735: 5731: 5727: 5721: 5718: 5715: 5712: 5708: 5702: 5698: 5694: 5689: 5686: 5683: 5679: 5673: 5669: 5664: 5657: 5654: 5648: 5642: 5639: 5634: 5617: 5614: 5613: 5612: 5600: 5597: 5594: 5591: 5588: 5585: 5581: 5578: 5572: 5568: 5564: 5561: 5539: 5535: 5531: 5509: 5505: 5501: 5496: 5492: 5471: 5468: 5465: 5461: 5458: 5430: 5427: 5397: 5394: 5371: 5368: 5351: 5337: 5334: 5331: 5328: 5325: 5322: 5310: 5309: 5297: 5275: 5272: 5268: 5245: 5242: 5227: 5226: 5215: 5212: 5207: 5203: 5199: 5196: 5193: 5188: 5185: 5182: 5178: 5172: 5168: 5164: 5159: 5155: 5129: 5126: 5122: 5110: 5109: 5098: 5095: 5090: 5087: 5083: 5079: 5074: 5070: 5066: 5063: 5060: 5055: 5052: 5049: 5045: 5039: 5035: 5031: 5026: 5022: 5018: 5004: 5003: 4992: 4989: 4984: 4981: 4977: 4971: 4967: 4963: 4960: 4957: 4952: 4949: 4945: 4939: 4936: 4933: 4929: 4923: 4919: 4915: 4910: 4907: 4903: 4897: 4893: 4880:, leading to 4867: 4864: 4860: 4856: 4853: 4842: 4841: 4830: 4827: 4824: 4819: 4815: 4811: 4808: 4805: 4797: 4794: 4791: 4787: 4783: 4778: 4773: 4770: 4767: 4763: 4754: 4750: 4746: 4738: 4734: 4730: 4725: 4720: 4716: 4683: 4680: 4676: 4663: 4662: 4651: 4648: 4643: 4640: 4636: 4630: 4626: 4622: 4619: 4616: 4611: 4608: 4604: 4598: 4595: 4592: 4588: 4582: 4578: 4574: 4569: 4566: 4562: 4556: 4552: 4539:, leading to 4526: 4523: 4519: 4515: 4512: 4501: 4500: 4489: 4486: 4483: 4478: 4474: 4470: 4467: 4464: 4456: 4453: 4450: 4446: 4442: 4437: 4432: 4429: 4426: 4422: 4413: 4409: 4405: 4397: 4393: 4389: 4384: 4379: 4375: 4348: 4326: 4323: 4319: 4298: 4295: 4279: 4278: 4273: 4260: 4255: 4252: 4212: 4208: 4185: 4181: 4139: 4138: 4137: 4136: 4125: 4122: 4119: 4113: 4107: 4103: 4099: 4096: 4093: 4090: 4087: 4084: 4081: 4076: 4072: 4068: 4065: 4062: 4057: 4053: 4049: 4044: 4040: 4036: 4033: 4030: 4024: 4008: 3993: 3990: 3987: 3982: 3979: 3976: 3973: 3970: 3967: 3964: 3960: 3957: 3951: 3948: 3945: 3941: 3938: 3932: 3927: 3923: 3919: 3916: 3913: 3902: 3891: 3888: 3885: 3879: 3873: 3869: 3863: 3859: 3855: 3850: 3846: 3842: 3839: 3834: 3830: 3826: 3821: 3817: 3813: 3810: 3807: 3801: 3785: 3770: 3767: 3764: 3757: 3753: 3749: 3745: 3742: 3736: 3733: 3729: 3726: 3720: 3715: 3711: 3707: 3704: 3701: 3675: 3672: 3669: 3664: 3660: 3656: 3652: 3649: 3645: 3642: 3638: 3635: 3631: 3626: 3622: 3618: 3615: 3612: 3596: 3595: 3594: 3593: 3582: 3579: 3576: 3570: 3564: 3560: 3556: 3553: 3550: 3547: 3544: 3541: 3538: 3533: 3529: 3525: 3522: 3519: 3514: 3510: 3506: 3501: 3497: 3493: 3490: 3487: 3483: 3467: 3456: 3453: 3450: 3447: 3444: 3439: 3435: 3431: 3428: 3425: 3422: 3419: 3416: 3413: 3409: 3405: 3402: 3399: 3394: 3390: 3383: 3380: 3376: 3371: 3366: 3362: 3358: 3355: 3352: 3348: 3341: 3338: 3334: 3306: 3303: 3300: 3297: 3294: 3291: 3288: 3285: 3282: 3279: 3275: 3272: 3268: 3265: 3262: 3258: 3255: 3251: 3246: 3242: 3238: 3235: 3232: 3216: 3215: 3214: 3213: 3200: 3197: 3194: 3188: 3182: 3178: 3172: 3169: 3164: 3160: 3154: 3151: 3148: 3145: 3142: 3139: 3134: 3130: 3124: 3120: 3104: 3091: 3088: 3085: 3080: 3077: 3073: 3070: 3064: 3061: 3058: 3055: 3052: 3049: 3045: 3042: 3036: 3008: 3003: 2998: 2995: 2990: 2987: 2984: 2981: 2978: 2975: 2971: 2968: 2962: 2959: 2956: 2953: 2950: 2947: 2943: 2940: 2934: 2920: 2919: 2918: 2917: 2906: 2903: 2900: 2896: 2890: 2886: 2882: 2879: 2876: 2871: 2867: 2862: 2858: 2855: 2852: 2849: 2846: 2843: 2840: 2837: 2834: 2831: 2827: 2823: 2818: 2814: 2810: 2807: 2804: 2801: 2798: 2795: 2791: 2775: 2764: 2761: 2758: 2755: 2752: 2749: 2743: 2740: 2735: 2732: 2725: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2697: 2694: 2690: 2686: 2678: 2674: 2670: 2665: 2660: 2656: 2649: 2646: 2643: 2640: 2637: 2634: 2607: 2603: 2600: 2597: 2594: 2591: 2588: 2583: 2580: 2577: 2574: 2571: 2568: 2565: 2561: 2558: 2553: 2549: 2546: 2543: 2540: 2537: 2534: 2531: 2528: 2525: 2522: 2518: 2514: 2510: 2507: 2503: 2500: 2497: 2494: 2491: 2488: 2472: 2471: 2470: 2469: 2458: 2455: 2452: 2449: 2446: 2442: 2439: 2435: 2432: 2429: 2425: 2422: 2410: 2397: 2394: 2391: 2385: 2379: 2375: 2371: 2368: 2365: 2360: 2356: 2352: 2349: 2346: 2341: 2337: 2332: 2316: 2305: 2302: 2299: 2296: 2292: 2289: 2285: 2282: 2278: 2275: 2271: 2268: 2265: 2262: 2259: 2232: 2228: 2225: 2220: 2217: 2214: 2211: 2208: 2205: 2202: 2198: 2195: 2191: 2188: 2185: 2181: 2178: 2161: 2160: 2159: 2158: 2143: 2140: 2137: 2134: 2130: 2124: 2120: 2116: 2113: 2108: 2104: 2100: 2097: 2094: 2089: 2085: 2079: 2075: 2070: 2054: 2041: 2038: 2035: 2032: 2029: 2023: 2020: 2015: 2012: 2006: 2003: 2000: 1992: 1988: 1984: 1979: 1974: 1970: 1961: 1957: 1945: 1934: 1931: 1928: 1925: 1922: 1919: 1914: 1910: 1906: 1903: 1900: 1897: 1894: 1891: 1886: 1883: 1880: 1877: 1874: 1870: 1864: 1861: 1858: 1854: 1848: 1845: 1842: 1838: 1834: 1831: 1828: 1825: 1820: 1817: 1814: 1810: 1804: 1800: 1773: 1770: 1767: 1764: 1761: 1756: 1753: 1750: 1746: 1740: 1736: 1732: 1729: 1726: 1723: 1720: 1715: 1711: 1705: 1700: 1697: 1694: 1690: 1675: 1674: 1673: 1672: 1657: 1654: 1651: 1648: 1644: 1638: 1634: 1630: 1625: 1621: 1617: 1612: 1608: 1604: 1601: 1596: 1592: 1588: 1585: 1580: 1576: 1570: 1566: 1561: 1545: 1534: 1531: 1528: 1525: 1520: 1516: 1512: 1507: 1503: 1499: 1496: 1490: 1487: 1482: 1479: 1473: 1470: 1462: 1458: 1454: 1449: 1444: 1440: 1431: 1427: 1399: 1395: 1392: 1387: 1384: 1381: 1378: 1375: 1370: 1366: 1362: 1357: 1353: 1349: 1346: 1342: 1339: 1335: 1332: 1328: 1325: 1319: 1315: 1298: 1297: 1296: 1295: 1283: 1259: 1256: 1253: 1250: 1246: 1240: 1236: 1232: 1229: 1226: 1221: 1217: 1212: 1196: 1181: 1178: 1175: 1172: 1169: 1166: 1162: 1159: 1132: 1129: 1126: 1123: 1120: 1117: 1113: 1110: 1088: 1085: 1066: 1065: 1052: 1049: 1046: 1043: 1040: 1037: 1034: 1031: 1028: 1025: 1022: 1019: 1016: 1011: 1007: 1001: 997: 993: 988: 984: 978: 974: 970: 965: 961: 955: 951: 947: 944: 941: 927: 926: 913: 910: 907: 904: 899: 895: 889: 885: 881: 878: 875: 872: 869: 864: 860: 854: 850: 846: 843: 840: 837: 834: 829: 825: 819: 815: 811: 808: 786: 785: 772: 767: 763: 759: 755: 752: 746: 742: 738: 734: 731: 725: 721: 717: 714: 687: 684: 680: 676: 651: 626: 603: 601: 600: 589: 586: 583: 580: 577: 574: 571: 568: 565: 562: 559: 556: 551: 547: 543: 540: 537: 534: 531: 528: 525: 522: 519: 514: 510: 506: 503: 500: 497: 494: 491: 488: 483: 480: 477: 473: 469: 466: 463: 458: 455: 452: 448: 444: 441: 438: 435: 432: 427: 423: 419: 416: 413: 408: 404: 325: 322: 319: 316: 313: 310: 307: 287: 284: 281: 278: 275: 272: 269: 266: 263: 237: 234: 233: 232: 204: 201: 170: 167: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 9006: 8995: 8992: 8990: 8987: 8985: 8982: 8980: 8977: 8975: 8972: 8970: 8967: 8965: 8962: 8960: 8957: 8956: 8954: 8944: 8937: 8933: 8929: 8925: 8921: 8917: 8913: 8909: 8906: 8903: 8901: 8897: 8893: 8889: 8885: 8884: 8883: 8882: 8878: 8874: 8871: 8867: 8863: 8859: 8848: 8844: 8840: 8836: 8832: 8828: 8827: 8826: 8823: 8820: 8814: 8813: 8807: 8803: 8799: 8795: 8791: 8787: 8786: 8785: 8784: 8780: 8776: 8752: 8748: 8744: 8738: 8732: 8724: 8720: 8716: 8710: 8707: 8702: 8698: 8694: 8688: 8682: 8674: 8670: 8666: 8658: 8654: 8650: 8644: 8638: 8630: 8626: 8622: 8617: 8613: 8609: 8606: 8603: 8600: 8595: 8592: 8588: 8584: 8581: 8576: 8572: 8568: 8563: 8559: 8555: 8552: 8549: 8542: 8541: 8540: 8524: 8521: 8517: 8513: 8505: 8500: 8497: 8494: 8490: 8466: 8463: 8460: 8455: 8452: 8448: 8444: 8441: 8436: 8432: 8428: 8423: 8419: 8415: 8412: 8409: 8402: 8401: 8400: 8383: 8378: 8375: 8371: 8367: 8364: 8360: 8354: 8351: 8347: 8343: 8339: 8332: 8329: 8325: 8315: 8314: 8313: 8307: 8303: 8299: 8295: 8287: 8286: 8285: 8284: 8280: 8276: 8268: 8265: 8261: 8258: 8255: 8251: 8250: 8249: 8246: 8243: 8239: 8230: 8227: 8226: 8225: 8224: 8220: 8216: 8215: 8209: 8206: 8205: 8204: 8203: 8199: 8195: 8194: 8188: 8187: 8186: 8185: 8181: 8177: 8173: 8170: 8169: 8163: 8160: 8157: 8156: 8155: 8154: 8150: 8146: 8145: 8139: 8136: 8133: 8130: 8129: 8128: 8127: 8124: 8120: 8119: 8118: 8115: 8113: 8105: 8103: 8102: 8098: 8094: 8090: 8086: 8082: 8074: 8070: 8066: 8061: 8060: 8057: 8053: 8049: 8044: 8043: 8042: 8041: 8037: 8033: 8028: 8026: 8020: 8018: 8009: 8007: 8006: 8002: 7998: 7994: 7986: 7984: 7983: 7979: 7975: 7969: 7967: 7966:combinatorics 7963: 7959: 7954: 7952: 7943: 7940: 7937: 7934: 7933: 7932: 7926: 7922: 7918: 7914: 7910: 7906: 7902: 7898: 7895: 7892: 7889: 7888: 7887: 7886: 7882: 7878: 7872: 7858: 7852: 7848: 7844: 7839: 7838: 7837: 7833: 7829: 7825: 7821: 7820: 7819: 7818: 7814: 7810: 7806: 7803:I’ve gone to 7799: 7795: 7791: 7787: 7786: 7777: 7773: 7769: 7765: 7762: 7758: 7754: 7750: 7734: 7731: 7728: 7725: 7718: 7717: 7704: 7701: 7698: 7690: 7674: 7671: 7668: 7665: 7662: 7655: 7654: 7640: 7620: 7600: 7591: 7576: 7569:for some set 7554: 7551: 7548: 7538: 7534: 7522: 7518: 7514: 7510: 7507: 7506: 7505: 7504: 7503: 7502: 7497: 7492: 7488: 7484: 7480: 7476: 7472: 7468: 7463: 7462: 7461: 7460: 7456: 7452: 7446: 7444: 7436: 7432: 7429: 7426: 7422: 7421: 7420: 7419: 7415: 7411: 7403: 7399: 7395: 7391: 7385: 7379: 7378: 7377: 7376: 7372: 7368: 7364: 7358: 7356: 7352: 7344: 7340: 7336: 7332: 7327: 7326:long articles 7323: 7319: 7315: 7314: 7310: 7306: 7303: 7301:Best regards. 7300: 7299: 7295: 7294: 7290: 7289: 7286: 7282: 7278: 7274: 7270: 7267: 7266: 7262: 7259: 7258: 7255: 7252: 7251: 7246: 7245: 7242: 7239: 7238: 7233: 7231: 7228: 7227: 7223: 7219: 7218: 7214: 7210: 7209: 7204: 7200: 7196: 7192: 7191: 7190: 7189: 7186: 7182: 7178: 7173: 7170: 7168: 7164: 7160: 7156: 7152: 7151:Support Merge 7148: 7146: 7143: 7140: 7135: 7131: 7130:Support Merge 7128: 7127: 7123: 7121: 7120: 7117: 7114: 7110: 7106: 7102: 7094: 7092: 7091: 7087: 7083: 7079: 7075: 7071: 7067: 7061: 7059: 7057: 7053: 7049: 7045: 7034: 7025: 7021: 7017: 7013: 7006: 7003: 6996: 6994: 6992: 6988: 6984: 6980: 6973: 6969: 6962: 6958: 6954: 6950: 6945: 6926: 6920: 6917: 6914: 6908: 6900: 6896: 6892: 6888: 6885: 6878: 6870: 6867: 6864: 6860: 6856: 6853: 6850: 6842: 6839: 6836: 6829: 6822: 6814: 6810: 6806: 6798: 6791: 6784: 6776: 6772: 6763: 6760: 6759: 6757: 6756:0-13-011290-9 6754: 6750: 6746: 6745: 6744: 6743: 6739: 6735: 6734:Ebony Jackson 6726: 6724: 6723: 6719: 6715: 6693: 6690: 6684: 6681: 6676: 6672: 6668: 6663: 6657: 6654: 6651: 6648: 6644: 6638: 6634: 6630: 6625: 6622: 6619: 6615: 6609: 6605: 6598: 6592: 6589: 6584: 6576: 6575: 6574: 6558: 6554: 6550: 6545: 6541: 6537: 6532: 6528: 6503: 6497: 6494: 6488: 6485: 6480: 6476: 6472: 6466: 6463: 6457: 6454: 6449: 6445: 6441: 6436: 6432: 6424: 6423: 6422: 6405: 6399: 6396: 6390: 6387: 6382: 6378: 6374: 6368: 6365: 6358: 6355: 6352: 6349: 6345: 6339: 6335: 6331: 6326: 6323: 6320: 6316: 6310: 6306: 6299: 6293: 6290: 6285: 6277: 6276: 6275: 6255: 6252: 6246: 6243: 6238: 6234: 6230: 6225: 6219: 6216: 6213: 6210: 6206: 6200: 6196: 6192: 6187: 6184: 6181: 6177: 6171: 6167: 6160: 6154: 6151: 6146: 6138: 6137: 6136: 6134: 6113: 6108: 6105: 6102: 6099: 6095: 6089: 6085: 6081: 6076: 6072: 6064: 6063: 6062: 6061: 6057: 6056: 6039: 6036: 6033: 6029: 6023: 6019: 6015: 6010: 6006: 5998: 5997: 5996: 5992: 5990: 5986: 5985: 5981: 5977: 5973: 5959: 5939: 5936: 5927: 5910: 5907: 5902: 5878: 5875: 5870: 5846: 5843: 5838: 5828: 5809: 5803: 5800: 5794: 5791: 5786: 5782: 5778: 5774: 5767: 5764: 5759: 5756: 5749: 5745: 5742: 5737: 5733: 5729: 5725: 5719: 5716: 5713: 5710: 5706: 5700: 5696: 5692: 5687: 5684: 5681: 5677: 5671: 5667: 5662: 5655: 5652: 5646: 5640: 5637: 5632: 5624: 5623: 5622: 5615: 5598: 5595: 5592: 5589: 5586: 5583: 5579: 5570: 5566: 5562: 5537: 5533: 5529: 5507: 5503: 5499: 5494: 5490: 5469: 5466: 5463: 5459: 5456: 5447: 5446: 5445: 5444: 5440: 5436: 5429:vector space? 5428: 5426: 5425: 5421: 5417: 5413: 5410: 5407: 5404: 5401: 5395: 5393: 5392: 5389: 5385: 5383: 5379: 5375: 5369: 5367: 5366: 5362: 5358: 5352: 5349: 5335: 5332: 5329: 5326: 5323: 5320: 5295: 5273: 5270: 5266: 5257: 5253: 5252: 5250: 5243: 5241: 5240: 5236: 5232: 5213: 5210: 5205: 5201: 5197: 5194: 5191: 5186: 5183: 5180: 5176: 5170: 5166: 5162: 5157: 5153: 5145: 5144: 5143: 5127: 5124: 5120: 5096: 5093: 5088: 5085: 5081: 5072: 5068: 5064: 5061: 5058: 5053: 5050: 5047: 5043: 5037: 5033: 5029: 5024: 5020: 5009: 5008: 5007: 4990: 4987: 4982: 4979: 4975: 4969: 4965: 4961: 4958: 4955: 4950: 4947: 4943: 4937: 4934: 4931: 4927: 4921: 4917: 4913: 4908: 4905: 4901: 4895: 4891: 4883: 4882: 4881: 4865: 4862: 4858: 4854: 4851: 4828: 4825: 4822: 4817: 4813: 4809: 4806: 4803: 4795: 4792: 4789: 4785: 4781: 4776: 4771: 4768: 4765: 4761: 4752: 4748: 4744: 4736: 4732: 4728: 4723: 4718: 4714: 4703: 4702: 4701: 4697: 4681: 4678: 4674: 4649: 4646: 4641: 4638: 4634: 4628: 4624: 4620: 4617: 4614: 4609: 4606: 4602: 4596: 4593: 4590: 4586: 4580: 4576: 4572: 4567: 4564: 4560: 4554: 4550: 4542: 4541: 4540: 4524: 4521: 4517: 4513: 4510: 4487: 4484: 4481: 4476: 4472: 4468: 4465: 4462: 4454: 4451: 4448: 4444: 4440: 4435: 4430: 4427: 4424: 4420: 4411: 4407: 4403: 4395: 4391: 4387: 4382: 4377: 4373: 4362: 4361: 4360: 4346: 4324: 4321: 4317: 4308: 4302: 4296: 4294: 4293: 4289: 4285: 4276: 4271: 4267: 4262: 4259: 4253: 4251: 4250: 4246: 4242: 4237: 4236: 4232: 4228: 4210: 4206: 4183: 4179: 4168: 4167: 4163: 4159: 4154: 4153: 4149: 4145: 4123: 4120: 4117: 4111: 4105: 4101: 4094: 4091: 4088: 4082: 4079: 4074: 4070: 4066: 4063: 4060: 4055: 4051: 4042: 4038: 4034: 4031: 4022: 4013: 4009: 3991: 3988: 3985: 3977: 3974: 3971: 3965: 3962: 3958: 3955: 3949: 3946: 3943: 3939: 3936: 3925: 3921: 3917: 3914: 3903: 3889: 3886: 3883: 3877: 3871: 3867: 3861: 3857: 3853: 3848: 3844: 3840: 3837: 3832: 3828: 3819: 3815: 3811: 3808: 3799: 3790: 3786: 3768: 3765: 3762: 3755: 3751: 3747: 3743: 3740: 3734: 3731: 3727: 3724: 3713: 3709: 3705: 3702: 3691: 3688: 3687: 3673: 3670: 3667: 3662: 3658: 3654: 3650: 3647: 3643: 3640: 3636: 3633: 3624: 3620: 3616: 3613: 3601: 3598: 3597: 3580: 3577: 3574: 3568: 3562: 3558: 3551: 3548: 3545: 3539: 3536: 3531: 3527: 3523: 3520: 3517: 3512: 3508: 3499: 3495: 3491: 3488: 3481: 3472: 3468: 3454: 3451: 3445: 3437: 3433: 3426: 3423: 3420: 3414: 3411: 3407: 3400: 3392: 3388: 3381: 3378: 3374: 3364: 3360: 3356: 3353: 3346: 3339: 3336: 3332: 3322: 3319: 3318: 3304: 3301: 3298: 3292: 3289: 3286: 3280: 3277: 3273: 3270: 3266: 3263: 3260: 3256: 3253: 3244: 3240: 3236: 3233: 3221: 3218: 3217: 3198: 3195: 3192: 3186: 3180: 3176: 3170: 3167: 3162: 3158: 3149: 3146: 3143: 3137: 3132: 3128: 3122: 3118: 3109: 3105: 3089: 3086: 3083: 3078: 3075: 3071: 3068: 3059: 3056: 3053: 3047: 3043: 3040: 3034: 3026: 3023: 3022: 3006: 2996: 2993: 2988: 2985: 2982: 2979: 2976: 2973: 2969: 2966: 2957: 2954: 2951: 2945: 2941: 2938: 2932: 2925: 2922: 2921: 2904: 2901: 2898: 2894: 2888: 2884: 2880: 2877: 2874: 2869: 2865: 2860: 2856: 2850: 2847: 2844: 2841: 2838: 2832: 2829: 2825: 2821: 2816: 2812: 2805: 2802: 2799: 2793: 2789: 2780: 2776: 2762: 2759: 2756: 2753: 2750: 2747: 2741: 2738: 2733: 2730: 2723: 2719: 2713: 2710: 2707: 2704: 2701: 2695: 2692: 2688: 2684: 2676: 2672: 2668: 2663: 2658: 2654: 2644: 2641: 2638: 2632: 2624: 2621: 2620: 2601: 2598: 2595: 2592: 2589: 2586: 2581: 2578: 2575: 2572: 2569: 2566: 2563: 2559: 2556: 2551: 2547: 2544: 2541: 2535: 2532: 2529: 2526: 2523: 2516: 2512: 2508: 2505: 2498: 2495: 2492: 2486: 2477: 2474: 2473: 2456: 2453: 2450: 2447: 2444: 2440: 2437: 2433: 2430: 2427: 2423: 2420: 2411: 2395: 2392: 2389: 2383: 2377: 2373: 2369: 2366: 2363: 2358: 2354: 2350: 2347: 2344: 2339: 2335: 2330: 2321: 2317: 2303: 2300: 2297: 2294: 2290: 2287: 2283: 2280: 2276: 2273: 2269: 2263: 2257: 2249: 2246: 2245: 2226: 2223: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2196: 2193: 2189: 2186: 2183: 2179: 2176: 2166: 2163: 2162: 2141: 2138: 2135: 2132: 2128: 2122: 2118: 2114: 2111: 2106: 2102: 2098: 2095: 2092: 2087: 2083: 2077: 2073: 2068: 2059: 2055: 2039: 2036: 2033: 2030: 2027: 2021: 2018: 2013: 2010: 2004: 2001: 1998: 1990: 1986: 1982: 1977: 1972: 1968: 1959: 1955: 1946: 1932: 1929: 1923: 1917: 1912: 1908: 1904: 1901: 1898: 1892: 1881: 1878: 1875: 1868: 1862: 1859: 1856: 1852: 1846: 1843: 1840: 1836: 1832: 1826: 1815: 1808: 1802: 1798: 1789: 1786: 1785: 1771: 1768: 1762: 1751: 1744: 1738: 1730: 1727: 1724: 1721: 1713: 1709: 1703: 1698: 1695: 1692: 1688: 1680: 1677: 1676: 1655: 1652: 1649: 1646: 1642: 1636: 1632: 1623: 1619: 1615: 1610: 1606: 1599: 1594: 1590: 1586: 1583: 1578: 1574: 1568: 1564: 1559: 1550: 1546: 1532: 1529: 1526: 1518: 1514: 1510: 1505: 1501: 1494: 1488: 1485: 1480: 1477: 1471: 1468: 1460: 1456: 1452: 1447: 1442: 1438: 1429: 1425: 1416: 1413: 1412: 1393: 1390: 1385: 1382: 1379: 1376: 1368: 1364: 1360: 1355: 1351: 1344: 1340: 1337: 1333: 1330: 1326: 1323: 1317: 1313: 1303: 1300: 1299: 1281: 1257: 1254: 1251: 1248: 1244: 1238: 1234: 1230: 1227: 1224: 1219: 1215: 1210: 1201: 1197: 1179: 1176: 1173: 1170: 1167: 1164: 1160: 1157: 1148: 1147:Airy function 1145: 1144: 1130: 1127: 1124: 1121: 1118: 1115: 1111: 1108: 1098: 1095: 1094: 1093: 1086: 1084: 1083: 1079: 1075: 1071: 1044: 1038: 1032: 1029: 1023: 1017: 1009: 1005: 999: 995: 991: 986: 982: 976: 972: 968: 963: 959: 953: 949: 942: 939: 932: 931: 930: 908: 902: 897: 893: 887: 883: 879: 873: 867: 862: 858: 852: 848: 844: 838: 832: 827: 823: 817: 813: 809: 806: 799: 798: 797: 795: 791: 770: 765: 761: 757: 753: 750: 744: 740: 736: 732: 729: 723: 719: 715: 712: 705: 704: 703: 701: 685: 682: 678: 674: 666: 662: 656: 654: 647: 643: 640: 635: 633: 629: 622: 619: 615: 611: 607: 584: 578: 575: 569: 563: 557: 549: 545: 541: 535: 529: 526: 520: 512: 508: 504: 501: 498: 492: 486: 481: 478: 475: 471: 464: 456: 453: 450: 446: 442: 436: 430: 425: 421: 414: 406: 402: 394: 393: 392: 390: 386: 382: 378: 372: 369: 368: 364: 360: 355: 353: 348: 347: 343: 339: 323: 320: 317: 314: 311: 308: 305: 285: 282: 279: 276: 273: 270: 267: 264: 261: 252: 251: 247: 243: 242:128.192.21.39 236:Too Technical 235: 231: 228: 227: 222: 221: 220: 219: 215: 211: 202: 200: 198: 194: 190: 186: 180: 176: 168: 153: 149: 148:High-priority 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 73:High‑priority 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 8942: 8923: 8869: 8855: 8830: 8824: 8818: 8815: 8789: 8772: 8503: 8501: 8495: 8492: 8488: 8481: 8398: 8311: 8272: 8247: 8244: 8240: 8236: 8218: 8197: 8179: 8148: 8122: 8116: 8109: 8077: 8065:Toby Bartels 8032:Toby Bartels 8029: 8024: 8021: 8016: 8013: 7990: 7970: 7955: 7947: 7930: 7873: 7862: 7802: 7775: 7757:vector space 7520: 7516: 7508: 7447: 7440: 7425:Arthur Rubin 7407: 7359: 7348: 7317: 7284: 7253: 7240: 7229: 7213:vote against 7212: 7171: 7154: 7150: 7139:Arthur Rubin 7133: 7129: 7113:Arthur Rubin 7098: 7080: 7077: 7072: 7069: 7065: 7042:— Preceding 7038: 7015: 7011: 7005: 6977:— Preceding 6974: 6970: 6966: 6748: 6730: 6711: 6519: 6420: 6273: 6130: 5994: 5987: 5974: 5928: 5829: 5825: 5620: 5432: 5414: 5411: 5408: 5405: 5402: 5399: 5386: 5373: 5353: 5350: 5312: 5247: 5228: 5111: 5005: 4843: 4698: 4665: 4502: 4304: 4300: 4281: 4265: 4257: 4238: 4169: 4155: 4140: 4011: 3788: 3470: 3107: 2778: 2319: 2057: 1548: 1199: 1090: 1069: 1067: 928: 793: 789: 787: 699: 664: 663:rather than 660: 658: 649: 645: 641: 638: 636: 631: 624: 620: 617: 613: 609: 605: 602: 388: 384: 376: 374: 370: 356: 349: 253: 239: 226:Geometry guy 224: 206: 189:69.153.6.144 172: 147: 107: 51:WikiProjects 34: 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1092:below). 614:Dy = y' 185:unsigned 7805:WT:MATH 7772:partial 7318:against 7149:strong 5416:Jez 006 4844:we set 4503:we set 4010:==: --> 3787:==: --> 3469:==: --> 3106:==: --> 2777:==: --> 2318:==: --> 2056:==: --> 1547:==: --> 1198:==: --> 210:CptBork 150:on the 41:C-class 8775:Gamren 8482:where 7809:Loraof 7451:Loraof 7428:(talk) 7331:EngiZe 7283:cite. 7142:(talk) 7116:(talk) 5522:, but 5388:Xeṭrov 5112:Since 612:(i.e. 604:where 47:scale. 8918:and 7877:Purgy 7790:Purgy 7367:Purgy 7316:I am 7195:Purgy 7172:Don't 7159:Purgy 6714:Salix 5256:Euler 4307:Euler 3904:and: 792:into 639:order 28:This 8932:talk 8896:talk 8892:JonH 8877:talk 8843:talk 8798:talk 8779:talk 8491:= ∫ 8298:talk 8279:talk 8174:and 8097:talk 8093:JonH 8069:talk 8052:talk 8036:talk 8001:talk 7978:talk 7964:and 7917:talk 7881:talk 7847:talk 7843:JonH 7832:talk 7813:talk 7794:talk 7511:are 7455:talk 7414:talk 7394:talk 7371:talk 7335:talk 7309:talk 7275:and 7199:talk 7181:talk 7163:talk 7134:more 7111:. — 7086:talk 7052:talk 6987:talk 6953:talk 6949:JonH 6753:ISBN 6738:talk 6718:talk 6274:and 5980:talk 5830:The 5439:talk 5420:talk 5406:To: 5361:talk 5235:talk 4288:talk 4270:ISBN 4245:talk 4231:talk 4198:and 4162:talk 4148:talk 1078:talk 610:d/dx 363:talk 342:talk 246:talk 214:talk 193:talk 142:High 8868:. 8790:any 8499:." 8290:= 0 8091:). 7953:). 7519:or 7465:it: 7380:Hi 7020:doi 6720:): 6682:cos 6486:sin 6455:cos 6388:sin 6244:cos 6058:and 5960:cos 5926:?) 5894:? 5792:sin 5743:cos 634:). 383:of 8955:: 8934:) 8914:, 8910:, 8898:) 8879:) 8845:) 8800:) 8781:) 8717:≡ 8593:− 8582:∫ 8522:− 8496:dx 8453:− 8442:∫ 8376:− 8352:− 8300:) 8281:) 8178:, 8099:) 8071:) 8054:) 8038:) 8003:) 7980:) 7919:) 7911:) 7883:) 7849:) 7834:) 7815:) 7796:) 7763:). 7552:∈ 7457:) 7416:) 7396:) 7373:) 7337:) 7329:-- 7201:) 7183:) 7165:) 7088:) 7054:) 7016:48 7014:. 6989:) 6955:) 6868:− 6854:⋯ 6840:− 6758:: 6740:) 6712:-- 6685:⁡ 6649:− 6489:⁡ 6458:⁡ 6391:⁡ 6350:− 6332:− 6247:⁡ 6211:− 6100:− 5982:) 5972:. 5795:⁡ 5746:⁡ 5711:− 5593:≠ 5441:) 5422:) 5363:) 5237:) 5214:0. 5195:⋯ 5184:− 5097:0. 5062:⋯ 5051:− 4991:0. 4959:⋯ 4935:− 4807:⋯ 4793:− 4769:− 4650:0. 4618:⋯ 4594:− 4466:⋯ 4452:− 4428:− 4290:) 4247:) 4233:) 4164:) 4150:) 4061:− 4035:− 4014:: 3944:− 3918:− 3838:− 3812:− 3791:: 3732:− 3706:− 3692:: 3686:. 3641:− 3617:− 3518:− 3492:− 3455:0. 3357:− 3317:. 3261:− 3237:− 3147:− 3057:− 3021:. 2997:∈ 2955:− 2905:0. 2875:− 2833:− 2803:− 2763:0. 2748:− 2696:− 2642:− 2619:. 2602:∈ 2599:γ 2593:β 2587:α 2570:β 2567:α 2548:γ 2545:− 2530:β 2524:α 2496:− 2454:λ 2448:− 2428:− 2345:− 2301:λ 2298:− 2281:− 2244:. 2227:∈ 2184:− 2060:: 1933:0. 1902:⋯ 1879:− 1860:− 1844:− 1784:. 1689:∑ 1620:α 1616:− 1551:: 1515:α 1511:− 1417:: 1411:. 1394:∈ 1361:− 1282:λ 1228:λ 1225:− 1202:: 1165:− 1149:: 1143:. 1119:λ 1116:− 1080:) 616:, 502:⋯ 479:− 454:− 365:) 344:) 248:) 216:) 195:) 8930:( 8894:( 8875:( 8841:( 8796:( 8777:( 8758:) 8753:2 8749:c 8745:+ 8742:) 8739:x 8736:( 8733:H 8730:( 8725:F 8721:e 8714:) 8711:c 8708:+ 8703:1 8699:c 8695:+ 8692:) 8689:x 8686:( 8683:H 8680:( 8675:F 8671:e 8667:= 8664:) 8659:1 8655:c 8651:+ 8648:) 8645:x 8642:( 8639:H 8636:( 8631:F 8627:e 8623:+ 8618:F 8614:e 8610:c 8607:= 8604:x 8601:d 8596:F 8589:e 8585:g 8577:F 8573:e 8569:+ 8564:F 8560:e 8556:c 8553:= 8550:y 8525:F 8518:e 8514:g 8504:c 8493:f 8489:F 8484:c 8467:, 8464:x 8461:d 8456:F 8449:e 8445:g 8437:F 8433:e 8429:+ 8424:F 8420:e 8416:c 8413:= 8410:y 8384:. 8379:F 8372:e 8368:g 8365:= 8361:) 8355:F 8348:e 8344:y 8340:( 8333:x 8330:d 8326:d 8316:" 8296:( 8277:( 8095:( 8067:( 8050:( 8034:( 8030:— 7999:( 7976:( 7915:( 7879:( 7845:( 7830:( 7811:( 7792:( 7735:0 7732:= 7729:y 7726:L 7705:0 7702:= 7699:f 7675:. 7672:f 7669:= 7666:y 7663:L 7641:f 7621:y 7601:L 7577:I 7555:I 7549:i 7545:) 7539:i 7535:y 7531:( 7453:( 7412:( 7392:( 7386:: 7382:@ 7369:( 7333:( 7307:( 7197:( 7179:( 7161:( 7084:( 7050:( 7026:. 7022:: 6985:( 6951:( 6930:) 6927:x 6924:( 6921:F 6918:= 6915:y 6912:) 6909:x 6906:( 6901:n 6897:P 6893:+ 6889:′ 6886:y 6882:) 6879:x 6876:( 6871:1 6865:n 6861:P 6857:+ 6851:+ 6846:) 6843:1 6837:n 6834:( 6830:y 6826:) 6823:x 6820:( 6815:1 6811:P 6807:+ 6802:) 6799:n 6796:( 6792:y 6788:) 6785:x 6782:( 6777:0 6773:P 6736:( 6716:( 6697:) 6694:x 6691:k 6688:( 6677:0 6673:C 6669:= 6664:2 6658:x 6655:k 6652:i 6645:e 6639:0 6635:C 6631:+ 6626:x 6623:k 6620:i 6616:e 6610:0 6606:C 6599:= 6593:′ 6590:0 6585:y 6559:0 6555:C 6551:= 6546:1 6542:A 6538:= 6533:0 6529:A 6504:. 6501:) 6498:x 6495:k 6492:( 6481:1 6477:C 6473:+ 6470:) 6467:x 6464:k 6461:( 6450:0 6446:C 6442:= 6437:H 6433:y 6406:. 6403:) 6400:x 6397:k 6394:( 6383:1 6379:C 6375:= 6369:i 6366:2 6359:x 6356:k 6353:i 6346:e 6340:1 6336:A 6327:x 6324:k 6321:i 6317:e 6311:0 6307:A 6300:= 6294:′ 6291:1 6286:y 6259:) 6256:x 6253:k 6250:( 6239:0 6235:C 6231:= 6226:2 6220:x 6217:k 6214:i 6207:e 6201:1 6197:A 6193:+ 6188:x 6185:k 6182:i 6178:e 6172:0 6168:A 6161:= 6155:′ 6152:0 6147:y 6114:. 6109:x 6106:k 6103:i 6096:e 6090:1 6086:A 6082:= 6077:1 6073:y 6040:x 6037:k 6034:i 6030:e 6024:0 6020:A 6016:= 6011:0 6007:y 5978:( 5940:i 5937:2 5911:″ 5908:0 5903:y 5879:′ 5876:1 5871:y 5847:′ 5844:0 5839:y 5810:. 5807:) 5804:x 5801:k 5798:( 5787:1 5783:C 5779:= 5775:) 5768:i 5765:2 5760:x 5757:k 5750:( 5738:0 5734:C 5730:= 5726:) 5720:x 5717:k 5714:i 5707:e 5701:1 5697:A 5693:+ 5688:x 5685:k 5682:i 5678:e 5672:0 5668:A 5663:( 5656:2 5653:1 5647:= 5641:′ 5638:0 5633:y 5611:. 5599:x 5596:2 5590:x 5587:4 5584:= 5580:′ 5577:) 5571:2 5567:x 5563:2 5560:( 5538:0 5534:y 5530:2 5508:2 5504:x 5500:= 5495:0 5491:y 5470:x 5467:2 5464:= 5460:′ 5457:y 5437:( 5418:( 5359:( 5336:y 5333:i 5330:+ 5327:x 5324:= 5321:z 5296:z 5274:x 5271:z 5267:e 5233:( 5211:= 5206:N 5202:A 5198:+ 5192:+ 5187:1 5181:N 5177:r 5171:1 5167:A 5163:+ 5158:N 5154:r 5128:t 5125:r 5121:e 5094:= 5089:t 5086:r 5082:e 5078:) 5073:N 5069:A 5065:+ 5059:+ 5054:1 5048:N 5044:r 5038:1 5034:A 5030:+ 5025:N 5021:r 5017:( 4988:= 4983:t 4980:r 4976:e 4970:N 4966:A 4962:+ 4956:+ 4951:t 4948:r 4944:e 4938:1 4932:N 4928:r 4922:1 4918:A 4914:+ 4909:t 4906:r 4902:e 4896:N 4892:r 4866:t 4863:r 4859:e 4855:= 4852:y 4829:0 4826:= 4823:y 4818:N 4814:A 4810:+ 4804:+ 4796:1 4790:N 4786:t 4782:d 4777:y 4772:1 4766:N 4762:d 4753:1 4749:A 4745:+ 4737:N 4733:t 4729:d 4724:y 4719:N 4715:d 4682:t 4679:r 4675:e 4647:= 4642:x 4639:z 4635:e 4629:n 4625:A 4621:+ 4615:+ 4610:x 4607:z 4603:e 4597:1 4591:n 4587:z 4581:1 4577:A 4573:+ 4568:x 4565:z 4561:e 4555:n 4551:z 4525:x 4522:z 4518:e 4514:= 4511:y 4488:0 4485:= 4482:y 4477:n 4473:A 4469:+ 4463:+ 4455:1 4449:n 4445:x 4441:d 4436:y 4431:1 4425:n 4421:d 4412:1 4408:A 4404:+ 4396:n 4392:x 4388:d 4383:y 4378:n 4374:d 4347:z 4325:x 4322:z 4318:e 4286:( 4243:( 4229:( 4211:0 4207:D 4184:1 4180:D 4160:( 4146:( 4124:0 4121:= 4118:y 4112:] 4106:0 4102:D 4098:) 4095:2 4092:+ 4089:n 4086:( 4083:n 4080:+ 4075:1 4071:D 4067:x 4064:3 4056:2 4052:D 4048:) 4043:2 4039:x 4032:1 4029:( 4023:[ 3992:0 3989:= 3986:y 3981:) 3978:2 3975:+ 3972:n 3969:( 3966:n 3963:+ 3959:′ 3956:y 3950:x 3947:3 3940:″ 3937:y 3931:) 3926:2 3922:x 3915:1 3912:( 3890:0 3887:= 3884:y 3878:] 3872:0 3868:D 3862:2 3858:n 3854:+ 3849:1 3845:D 3841:x 3833:2 3829:D 3825:) 3820:2 3816:x 3809:1 3806:( 3800:[ 3769:0 3766:= 3763:y 3756:2 3752:n 3748:+ 3744:′ 3741:y 3735:x 3728:″ 3725:y 3719:) 3714:2 3710:x 3703:1 3700:( 3674:0 3671:= 3668:y 3663:2 3659:n 3655:+ 3651:′ 3648:y 3644:x 3637:″ 3634:y 3630:) 3625:2 3621:x 3614:1 3611:( 3581:0 3578:= 3575:y 3569:] 3563:0 3559:D 3555:) 3552:1 3549:+ 3546:n 3543:( 3540:n 3537:+ 3532:1 3528:D 3524:x 3521:2 3513:2 3509:D 3505:) 3500:2 3496:x 3489:1 3486:( 3482:[ 3452:= 3449:) 3446:x 3443:( 3438:n 3434:P 3430:) 3427:1 3424:+ 3421:n 3418:( 3415:n 3412:+ 3408:] 3404:) 3401:x 3398:( 3393:n 3389:P 3382:x 3379:d 3375:d 3370:) 3365:2 3361:x 3354:1 3351:( 3347:[ 3340:x 3337:d 3333:d 3323:: 3305:0 3302:= 3299:y 3296:) 3293:1 3290:+ 3287:n 3284:( 3281:n 3278:+ 3274:′ 3271:y 3267:x 3264:2 3257:″ 3254:y 3250:) 3245:2 3241:x 3234:1 3231:( 3199:0 3196:= 3193:y 3187:] 3181:0 3177:D 3171:n 3168:+ 3163:1 3159:D 3153:) 3150:x 3144:1 3141:( 3138:+ 3133:2 3129:D 3123:x 3119:[ 3110:: 3090:0 3087:= 3084:y 3079:n 3076:+ 3072:′ 3069:y 3063:) 3060:x 3054:1 3051:( 3048:+ 3044:″ 3041:y 3035:x 3027:: 3007:0 3002:N 2994:n 2989:, 2986:0 2983:= 2980:y 2977:n 2974:+ 2970:′ 2967:y 2961:) 2958:x 2952:1 2949:( 2946:+ 2942:″ 2939:y 2933:x 2902:= 2899:w 2895:] 2889:0 2885:D 2881:b 2878:a 2870:1 2866:D 2861:) 2857:z 2854:) 2851:1 2848:+ 2845:b 2842:+ 2839:a 2836:( 2830:c 2826:( 2822:+ 2817:2 2813:D 2809:) 2806:z 2800:1 2797:( 2794:z 2790:[ 2760:= 2757:w 2754:b 2751:a 2742:z 2739:d 2734:w 2731:d 2724:] 2720:z 2717:) 2714:1 2711:+ 2708:b 2705:+ 2702:a 2699:( 2693:c 2689:[ 2685:+ 2677:2 2673:z 2669:d 2664:w 2659:2 2655:d 2648:) 2645:z 2639:1 2636:( 2633:z 2625:: 2606:R 2596:, 2590:, 2582:, 2579:0 2576:= 2573:y 2564:+ 2560:′ 2557:y 2552:) 2542:x 2539:) 2536:1 2533:+ 2527:+ 2521:( 2517:( 2513:+ 2509:″ 2506:y 2502:) 2499:1 2493:x 2490:( 2487:x 2457:u 2451:2 2445:= 2441:′ 2438:u 2434:x 2431:2 2424:″ 2421:u 2396:0 2393:= 2390:y 2384:] 2378:0 2374:D 2370:n 2367:2 2364:+ 2359:1 2355:D 2351:x 2348:2 2340:2 2336:D 2331:[ 2304:u 2295:= 2291:′ 2288:u 2284:x 2277:″ 2274:u 2270:= 2267:] 2264:u 2261:[ 2258:L 2250:: 2231:Z 2224:n 2219:, 2216:0 2213:= 2210:y 2207:n 2204:2 2201:+ 2197:′ 2194:y 2190:x 2187:2 2180:″ 2177:y 2142:, 2139:0 2136:= 2133:y 2129:] 2123:0 2119:D 2115:b 2112:+ 2107:1 2103:D 2099:x 2096:a 2093:+ 2088:2 2084:D 2078:2 2074:x 2069:[ 2040:0 2037:= 2034:y 2031:b 2028:+ 2022:x 2019:d 2014:y 2011:d 2005:x 2002:a 1999:+ 1991:2 1987:x 1983:d 1978:y 1973:2 1969:d 1960:2 1956:x 1930:= 1927:) 1924:x 1921:( 1918:y 1913:0 1909:a 1905:+ 1899:+ 1896:) 1893:x 1890:( 1885:) 1882:1 1876:n 1873:( 1869:y 1863:1 1857:n 1853:x 1847:1 1841:n 1837:a 1833:+ 1830:) 1827:x 1824:( 1819:) 1816:n 1813:( 1809:y 1803:n 1799:x 1790:: 1772:0 1769:= 1766:) 1763:x 1760:( 1755:) 1752:i 1749:( 1745:y 1739:i 1735:) 1731:d 1728:+ 1725:x 1722:c 1719:( 1714:i 1710:b 1704:n 1699:0 1696:= 1693:i 1656:, 1653:0 1650:= 1647:y 1643:] 1637:0 1633:D 1629:) 1624:2 1611:2 1607:x 1603:( 1600:+ 1595:1 1591:D 1587:x 1584:+ 1579:2 1575:D 1569:2 1565:x 1560:[ 1533:0 1530:= 1527:y 1524:) 1519:2 1506:2 1502:x 1498:( 1495:+ 1489:x 1486:d 1481:y 1478:d 1472:x 1469:+ 1461:2 1457:x 1453:d 1448:y 1443:2 1439:d 1430:2 1426:x 1398:R 1391:n 1386:, 1383:0 1380:= 1377:y 1374:) 1369:2 1365:n 1356:2 1352:x 1348:( 1345:+ 1341:′ 1338:y 1334:x 1331:+ 1327:″ 1324:y 1318:2 1314:x 1258:, 1255:0 1252:= 1249:y 1245:] 1239:0 1235:D 1231:x 1220:2 1216:D 1211:[ 1180:, 1177:0 1174:= 1171:y 1168:x 1161:″ 1158:y 1131:0 1128:= 1125:y 1122:x 1112:″ 1109:y 1076:( 1070:D 1051:] 1048:) 1045:x 1042:( 1039:f 1036:[ 1033:L 1030:= 1027:) 1024:x 1021:( 1018:f 1015:] 1010:0 1006:D 1000:2 996:p 992:+ 987:1 983:D 977:1 973:p 969:+ 964:2 960:D 954:0 950:p 946:[ 943:= 940:g 912:) 909:x 906:( 903:f 898:0 894:D 888:2 884:p 880:+ 877:) 874:x 871:( 868:f 863:1 859:D 853:1 849:p 845:+ 842:) 839:x 836:( 833:f 828:2 824:D 818:0 814:p 810:= 807:g 794:g 790:f 771:f 766:2 762:p 758:+ 754:′ 751:f 745:1 741:p 737:+ 733:″ 730:f 724:0 720:p 716:= 713:g 700:D 686:t 683:d 679:/ 675:d 665:x 661:t 652:n 650:a 646:y 642:n 632:t 627:i 625:a 618:D 606:D 588:) 585:x 582:( 579:f 576:= 573:) 570:x 567:( 564:y 561:) 558:x 555:( 550:0 546:a 542:+ 539:) 536:x 533:( 530:y 527:D 524:) 521:x 518:( 513:1 509:a 505:+ 499:+ 496:) 493:x 490:( 487:y 482:1 476:n 472:D 468:) 465:x 462:( 457:1 451:n 447:a 443:+ 440:) 437:x 434:( 431:y 426:n 422:D 418:) 415:x 412:( 407:n 403:a 389:y 385:y 377:L 361:( 340:( 324:0 321:= 318:y 315:b 312:+ 309:x 306:a 286:0 283:= 280:c 277:+ 274:y 271:b 268:+ 265:x 262:a 244:( 212:( 191:( 154:. 53::

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