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I moved a lot of material (mostly examples) from ordinary differential equations to this article. I tried to integrate the material a bit but was not very successful. This article seriously needs a complete rewrite. MathMartin 18:21, 18 March 2007 (UTC) Thanks —Preceding unsigned comment added by 69.153.6.144 ( talk) 02:38, 16 February 2008 (UTC)
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 were unaware that this article only applies to ODE's, not PDE's. Otherwise I think the content looks pretty solid. CptBork ( talk) 19:19, 13 June 2008 (UTC)
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. 128.192.21.39 ( talk) 15:29, 26 September 2008 (UTC)
I must confess... that I agree with you. A linear differential equation is "just like" a line, but a line in general form. So! is a good starting position for a line in 2D. And, is the homogeneous form. I will develop this theme on paper for a bit... — Михал Орела ( talk) 09:36, 15 September 2009 (UTC)
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 Shroud of Turin as a practical illustration in his book. And so, I have linked the math to a Wikipedia article on the subject.
Now I will look for some more interesting simple examples... such as electric circuits,... — Михал Орела ( talk) 11:49, 15 September 2009 (UTC)
I had commented out the following text in the original article
The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation would be
where D is the differential operator d/dx (i.e. Dy = y' , D2y = y",... ), and the ai are given functions. and the source term is considered to be a function of time ƒ(t).Such an equation is said to have order n, the index of the highest derivative of y that is involved. (Assuming a possibly existing coefficient an 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.)
Now that I have tried to edit the introduction in terms of variable t rather than x and used conventional differential forms I am beginning to think that the D 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
to explain the linear transformation of the function f into g. So let us compare with
and then with a little rewriting we have
So! I think I will try to re-introduce the D notation as illustrated above. It is important precisely because it is already used in the examples later on in the article. — Михал Орела ( talk) 17:05, 16 September 2009 (UTC)
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 Wikipedia. Birkoff and Rota introduce the subject of second order linear differential equations with the Bessel differential equation (number 2 in the list below).
I will check each of these in the English Wikipedia (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). — Михал Орела ( talk) 17:41, 16 September 2009 (UTC)
Taking a break :) — Михал Орела ( talk) 18:50, 16 September 2009 (UTC)
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 and to make sure that no errors were made. — Михал Орела ( talk) 20:14, 16 September 2009 (UTC)
The next stage is to use uniform notation for all equations (where possible) and to cite sources (other than the Wikipedias). — Михал Орела ( talk) 20:14, 16 September 2009 (UTC)
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:
Then I will add in other reference works as appropriate. — Михал Орела ( talk) 13:12, 14 September 2009 (UTC)
Now I want to tidy up the following:
The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form , for possibly-complex values of . 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
we set , leading to
Specifically, for consistency with the introductory text it is more appropriate to use the exponential as a function of time.
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
we set , leading to
and this factors as
Since can not be zero then we have the classic characteristic equation:
So! This is what I propose to do next. — Михал Орела ( talk) 14:33, 14 September 2009 (UTC)
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 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.
- The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form , for possibly-complex values of . 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.
I find this use to have strange look in the context. In particular how shall we write z? Is it ? Not is this context!
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). — Михал Орела ( talk) 08:25, 15 September 2009 (UTC)
[1] shows that, back in 2006, someone added "_{0 \choose f}" to the last equation before the example subsection in section Linear_differential_equation#Nonhomogeneous_equation_with_constant_coefficients (the edit is from when the section was part of Ordinary_differential_equation). 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!
Xeṭrov 07:16, 18 November 2010 (UTC)
I hope no one minds that I'm changing the first sentence from:
"In mathematics, a linear differential equation is of the form:"
To:
"Linear differential equations are of the form:"
This clearly falls under the subject of mathematics, and even if it somehow is not then linear differential equations are still of that form...
Jez 006 ( talk) 17:09, 11 May 2011 (UTC)
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.-- Sandrobt ( talk) 05:11, 9 January 2013 (UTC)
In particular, the following solutions can be constructed
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.
The 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. (? ?)
I don't think the cosine form should have a denominator of in the argument of .
Jmichael ll ( talk) 02:37, 22 May 2013 (UTC)
Well spotted it was introduced in this edit in February [2]. I've reverted it. The current text is
The solutions are, respectively,
- and
These solutions provide a basis for the two-dimensional solution space 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
and
These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:
which is still not perfect as we need for equality to hold. It may be better to write
-- Salix ( talk): 04:28, 22 May 2013 (UTC)
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 article to reflect this. Ebony Jackson ( talk) 16:04, 30 December 2013 (UTC)
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? Should the lower integration limit perhaps be zero rather than a? Help please.
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. [1]
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 ... — Preceding unsigned comment added by 128.253.104.13 ( talk) 13:55, 16 September 2022 (UTC)
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. Potentially there may be a neater/more elegant way to write this though, as the paragraph is already quite bracket heavy. — Preceding unsigned comment added by 2.101.31.42 ( talk) 10:34, 19 November 2015 (UTC)
The first paragraph of the "Introduction" reads
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.
Vancan1ty ( talk) 17:35, 1 May 2016 (UTC)
I propose that Exponential response formula be merged as at most one paragraph of explanation to Linear differential equation#Exponential response formula. See also previous discussion at Talk:Exponential response formula. — Arthur Rubin (talk) 05:39, 30 May 2017 (UTC)
There is a lot of heavy pseudo-activity in editing the article to be merged 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 proposed merge. Perhaps the ~150 single(!) IP-edits should be checked against the involved editor(s).
I'll put this note also in the
other article's talk page, but -at my discretion- stop commenting on these matters.
Purgy (
talk) 07:49, 3 June 2017 (UTC)
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. Wandalen ( talk) 13:39, 3 June 2017 (UTC)
I’ve reverted Purgy Purgatorio’s major revamp of the lead. Given the substantial extent of the reverted changes, the rewrite needs to be discussed here.
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. Loraof ( talk) 21:00, 26 January 2018 (UTC)
I’ve gone to WT:MATH and requested participation in this thread. Loraof ( talk) 17:20, 27 January 2018 (UTC)
- 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.
- Maybe the b(x) should be moved to the LHS of the equation for this polynomial view on the LDEQ.
- 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.
- The notion of the "order" of a DEQ should be mentioned within the lead.
- 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.
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. Purgy ( talk) 10:56, 28 January 2018 (UTC)
I have rewritten the whole article in the spirit of the new lead. The objective was
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 above discussion).
Finally, I have added a section on holonomic functions. 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 calculus and combinatorics, and making algorithmic many operations of calculus (such as antiderivative), for which there was only heuristics with the standard definition and representation of functions.
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. D.Lazard ( talk) 17:44, 6 February 2018 (UTC)
IMHO, with the new version of the article, the rating should be upgraded to WP:B-class 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. D.Lazard ( talk) 07:34, 28 February 2018 (UTC)
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 continuous. 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.
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. For this reason, I hesitate to simply change every appearance of differentiable coefficients into continuous coefficients. (And of course, the solution still has to be differentiable!) But for those who know more, would that be a correct change to make?
— Toby Bartels ( talk) 19:54, 20 July 2018 (UTC)
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 example, in an electrical problem it could be a square wave function. It is said that an advantage of the method of Laplace transform applied to differential equations is that it can be applied to discontinuous functions (see Introduction to Laplace Transforms for Engineers). JonH ( talk) 10:20, 1 September 2018 (UTC)
This article currently uses the term "characteristic equation" once and the term "characteristic polynomial" 11 times. But in the article Characteristic polynomial, 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.
In the gigantic literature on linear differential equations, I checked the use of these two terms in the following references:
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.
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.
What do we want to do to clarify the terminology for students?
I see four options:
What do you prefer? Is there a clear majority among the editors of this article for one option or another? J.P. Martin-Flatin ( talk) 11:25, 13 July 2020 (UTC)
Currently the section "First-order equation with variable coefficients" concludes
Thus, the general solution is
where c is a constant of integration, and F = ∫ f dx."
It seems to me that c is already contained in the integral. If we say that H is an antiderivative of , we get
And then the constant simply gets absorbed into the integral.__ Gamren ( talk) 11:01, 12 February 2021 (UTC)
There has been a new article created recently on the
Armour formula, 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
Armour formula just doubles what is already on this page, in the section
First-order equation with variable coefficients.
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.
Felix QW (
talk) 18:55, 6 January 2022 (UTC)
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I moved a lot of material (mostly examples) from ordinary differential equations to this article. I tried to integrate the material a bit but was not very successful. This article seriously needs a complete rewrite. MathMartin 18:21, 18 March 2007 (UTC) Thanks —Preceding unsigned comment added by 69.153.6.144 ( talk) 02:38, 16 February 2008 (UTC)
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 were unaware that this article only applies to ODE's, not PDE's. Otherwise I think the content looks pretty solid. CptBork ( talk) 19:19, 13 June 2008 (UTC)
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. 128.192.21.39 ( talk) 15:29, 26 September 2008 (UTC)
I must confess... that I agree with you. A linear differential equation is "just like" a line, but a line in general form. So! is a good starting position for a line in 2D. And, is the homogeneous form. I will develop this theme on paper for a bit... — Михал Орела ( talk) 09:36, 15 September 2009 (UTC)
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 Shroud of Turin as a practical illustration in his book. And so, I have linked the math to a Wikipedia article on the subject.
Now I will look for some more interesting simple examples... such as electric circuits,... — Михал Орела ( talk) 11:49, 15 September 2009 (UTC)
I had commented out the following text in the original article
The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation would be
where D is the differential operator d/dx (i.e. Dy = y' , D2y = y",... ), and the ai are given functions. and the source term is considered to be a function of time ƒ(t).Such an equation is said to have order n, the index of the highest derivative of y that is involved. (Assuming a possibly existing coefficient an 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.)
Now that I have tried to edit the introduction in terms of variable t rather than x and used conventional differential forms I am beginning to think that the D 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
to explain the linear transformation of the function f into g. So let us compare with
and then with a little rewriting we have
So! I think I will try to re-introduce the D notation as illustrated above. It is important precisely because it is already used in the examples later on in the article. — Михал Орела ( talk) 17:05, 16 September 2009 (UTC)
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 Wikipedia. Birkoff and Rota introduce the subject of second order linear differential equations with the Bessel differential equation (number 2 in the list below).
I will check each of these in the English Wikipedia (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). — Михал Орела ( talk) 17:41, 16 September 2009 (UTC)
Taking a break :) — Михал Орела ( talk) 18:50, 16 September 2009 (UTC)
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 and to make sure that no errors were made. — Михал Орела ( talk) 20:14, 16 September 2009 (UTC)
The next stage is to use uniform notation for all equations (where possible) and to cite sources (other than the Wikipedias). — Михал Орела ( talk) 20:14, 16 September 2009 (UTC)
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:
Then I will add in other reference works as appropriate. — Михал Орела ( talk) 13:12, 14 September 2009 (UTC)
Now I want to tidy up the following:
The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form , for possibly-complex values of . 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
we set , leading to
Specifically, for consistency with the introductory text it is more appropriate to use the exponential as a function of time.
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
we set , leading to
and this factors as
Since can not be zero then we have the classic characteristic equation:
So! This is what I propose to do next. — Михал Орела ( talk) 14:33, 14 September 2009 (UTC)
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 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.
- The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form , for possibly-complex values of . 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.
I find this use to have strange look in the context. In particular how shall we write z? Is it ? Not is this context!
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). — Михал Орела ( talk) 08:25, 15 September 2009 (UTC)
[1] shows that, back in 2006, someone added "_{0 \choose f}" to the last equation before the example subsection in section Linear_differential_equation#Nonhomogeneous_equation_with_constant_coefficients (the edit is from when the section was part of Ordinary_differential_equation). 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!
Xeṭrov 07:16, 18 November 2010 (UTC)
I hope no one minds that I'm changing the first sentence from:
"In mathematics, a linear differential equation is of the form:"
To:
"Linear differential equations are of the form:"
This clearly falls under the subject of mathematics, and even if it somehow is not then linear differential equations are still of that form...
Jez 006 ( talk) 17:09, 11 May 2011 (UTC)
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.-- Sandrobt ( talk) 05:11, 9 January 2013 (UTC)
In particular, the following solutions can be constructed
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.
The 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. (? ?)
I don't think the cosine form should have a denominator of in the argument of .
Jmichael ll ( talk) 02:37, 22 May 2013 (UTC)
Well spotted it was introduced in this edit in February [2]. I've reverted it. The current text is
The solutions are, respectively,
- and
These solutions provide a basis for the two-dimensional solution space 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
and
These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:
which is still not perfect as we need for equality to hold. It may be better to write
-- Salix ( talk): 04:28, 22 May 2013 (UTC)
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 article to reflect this. Ebony Jackson ( talk) 16:04, 30 December 2013 (UTC)
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? Should the lower integration limit perhaps be zero rather than a? Help please.
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. [1]
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 ... — Preceding unsigned comment added by 128.253.104.13 ( talk) 13:55, 16 September 2022 (UTC)
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. Potentially there may be a neater/more elegant way to write this though, as the paragraph is already quite bracket heavy. — Preceding unsigned comment added by 2.101.31.42 ( talk) 10:34, 19 November 2015 (UTC)
The first paragraph of the "Introduction" reads
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.
Vancan1ty ( talk) 17:35, 1 May 2016 (UTC)
I propose that Exponential response formula be merged as at most one paragraph of explanation to Linear differential equation#Exponential response formula. See also previous discussion at Talk:Exponential response formula. — Arthur Rubin (talk) 05:39, 30 May 2017 (UTC)
There is a lot of heavy pseudo-activity in editing the article to be merged 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 proposed merge. Perhaps the ~150 single(!) IP-edits should be checked against the involved editor(s).
I'll put this note also in the
other article's talk page, but -at my discretion- stop commenting on these matters.
Purgy (
talk) 07:49, 3 June 2017 (UTC)
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. Wandalen ( talk) 13:39, 3 June 2017 (UTC)
I’ve reverted Purgy Purgatorio’s major revamp of the lead. Given the substantial extent of the reverted changes, the rewrite needs to be discussed here.
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. Loraof ( talk) 21:00, 26 January 2018 (UTC)
I’ve gone to WT:MATH and requested participation in this thread. Loraof ( talk) 17:20, 27 January 2018 (UTC)
- 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.
- Maybe the b(x) should be moved to the LHS of the equation for this polynomial view on the LDEQ.
- 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.
- The notion of the "order" of a DEQ should be mentioned within the lead.
- 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.
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. Purgy ( talk) 10:56, 28 January 2018 (UTC)
I have rewritten the whole article in the spirit of the new lead. The objective was
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 above discussion).
Finally, I have added a section on holonomic functions. 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 calculus and combinatorics, and making algorithmic many operations of calculus (such as antiderivative), for which there was only heuristics with the standard definition and representation of functions.
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. D.Lazard ( talk) 17:44, 6 February 2018 (UTC)
IMHO, with the new version of the article, the rating should be upgraded to WP:B-class 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. D.Lazard ( talk) 07:34, 28 February 2018 (UTC)
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 continuous. 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.
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. For this reason, I hesitate to simply change every appearance of differentiable coefficients into continuous coefficients. (And of course, the solution still has to be differentiable!) But for those who know more, would that be a correct change to make?
— Toby Bartels ( talk) 19:54, 20 July 2018 (UTC)
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 example, in an electrical problem it could be a square wave function. It is said that an advantage of the method of Laplace transform applied to differential equations is that it can be applied to discontinuous functions (see Introduction to Laplace Transforms for Engineers). JonH ( talk) 10:20, 1 September 2018 (UTC)
This article currently uses the term "characteristic equation" once and the term "characteristic polynomial" 11 times. But in the article Characteristic polynomial, 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.
In the gigantic literature on linear differential equations, I checked the use of these two terms in the following references:
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.
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.
What do we want to do to clarify the terminology for students?
I see four options:
What do you prefer? Is there a clear majority among the editors of this article for one option or another? J.P. Martin-Flatin ( talk) 11:25, 13 July 2020 (UTC)
Currently the section "First-order equation with variable coefficients" concludes
Thus, the general solution is
where c is a constant of integration, and F = ∫ f dx."
It seems to me that c is already contained in the integral. If we say that H is an antiderivative of , we get
And then the constant simply gets absorbed into the integral.__ Gamren ( talk) 11:01, 12 February 2021 (UTC)
There has been a new article created recently on the
Armour formula, 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
Armour formula just doubles what is already on this page, in the section
First-order equation with variable coefficients.
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.
Felix QW (
talk) 18:55, 6 January 2022 (UTC)