Talk:Linear differential equation

message: 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)
thread_title:  Moved material from ordinary differential equations 

message: 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)
replies: text: The lead refers to both ODEs and PDEs. The solution is not to rename but to improve the article in my view. Geometry guy 21:17, 15 June 2008 (UTC)
thread_title:  Article Should be Re-titled: "Linear Ordinary Differential Equations" 

message: 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).
* Airysche Differentialgleichung \ y - \lambda xy = 0.
** Airy function : y - xy = 0 , \,\! 
** ==> Linear operator form: \left[ D^2 - \lambda xD^0 \right]y = 0 , \,\! (notice the inclusion of the \lambda parameter).
* Besselsche Differentialgleichung \ x^2 y + x y' + (x^2 - n^2) y = 0,\ n \in \mathbb{R}.
** Bessel function  : x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0
** ==>Linear operator form: \left[ x^2 D^2 + x D^1 +(x^2 - \alpha^2)D^0 \right] y = 0 , \,\!
* Eulersche Differentialgleichung \sum_{i=0}^n b_i(cx+d)^i y^{(i)}(x) = 0.
** Euler-Cauchy equation :x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \cdots + a_0 y(x) = 0.
** and also the form :x^2\frac{d^2y}{dx^2} + ax\frac{dy}{dx} + by = 0 \,
** ==> Linear operator form: \left[ x^2D^2 + ax D^1 +bD^0 \right]y = 0 , \,\!
* Hermitesche Differentialgleichung \ y - 2xy' + 2ny = 0,\ n \in \mathbb{Z}.
** Hermitian polynomials :L[u] = u - x u' = -\lambda u
** ==> Linear operator form (from the equation on German site): \left[ D^2 - 2 x D^1 + 2 n D^0\right]\, y = 0\!
** and also :u - 2xu'=-2\lambda u
* Hypergeometrische Differentialgleichung \ x(x - 1)y + \left((\alpha + \beta + 1)x - \gamma\right)y' + \alpha\beta y = 0,\ \alpha, \beta, \gamma \in \mathbb{R}.
** Hypergeometric differential equation :z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - abw = 0.
** ==> Linear operator form(English version) : \left[ z(1-z)D^2 + \left(c-(a+b+1)z \right)D^1  - ab D^0\right] w = 0.
* Laguerresche Differentialgleichung x \, y + (1-x)\,y' + n y = 0,\ n \in \mathbb{N}_0.
** Laguerre polynomials : x\,y + (1 - x)\,y' + n\,y = 0\, 
** ==> Linear operator form : \left[ x\,D^2 + (1 - x)\,D^1 + n\,D^0\right]\, y = 0\, 
* Legendresche Differentialgleichung \ (1-x^2)y - 2xy' + n(n+1)y = 0.
** Legendre polynomials :{d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.
** ==> Linear operator form (German site) : \left[ (1-x^2)D^2 -2x D^1 +n(n+1) D^0\right]\, y = 0 
* Tschebyschowsche Differentialgleichung \ (1-x^2)y - xy' + n^2y = 0.
** Chebyshev polynomials: (1-x^2)\,y - x\,y' + n^2\,y = 0 \,\!
** ==> Linear operator form : \left[\, (1-x^2)D^2 -x D^1 +n^2 D^0 \right]\, y = 0
** and: (1-x^2)\,y - 3x\,y' + n(n+2)\,y = 0 \,\!
** ==> Linear operator form : \left[\, (1-x^2)D^2 -3x D^1 + n(n+2) D^0 \right]\, y = 0
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)
thread_title:  Classical examples 

message: 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)
thread_title:  References 

message: 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 e^{z x}, for possibly-complex values of z. 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 
:\frac {d^{n}y} {dx^{n}} + A_{1}\frac {d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = 0
we set y=e^{z x}, leading to 
:z^n e^{zx} + A_1 z^{n-1} e^{zx} + \cdots + A_n e^{zx} = 0.
Specifically, for consistency with the introductory text it is more appropriate to use the exponential e^{r t} 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
:\frac {d^{N}y} {dt^{N}} + A_{1}\frac {d^{N-1}y} {dt^{N-1}} + \cdots + A_{N}y = 0
we set y=e^{r t}, leading to 
:r^N e^{rt} + A_1 r^{N-1} e^{rt} + \cdots + A_N e^{rt} = 0.
and this factors as
:(r^N  + A_1 r^{N-1}  + \cdots + A_N) e^{rt} = 0.
Since e^{r t} can not be zero then we have the classic characteristic equation:
:r^N  + A_1 r^{N-1}  + \cdots + A_N = 0.
So! This is what I propose to do next. —Михал Орела (talk) 14:33, 14 September 2009 (UTC)
replies: text: The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form e^{z x}, for possibly-complex values of z. 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 
z = x +iy? 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)
thread_title: Homogeneous equations with constant coefficients

message: 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 e^{z x}, for possibly-complex values of z. 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 
z = x +iy? 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)
thread_title:  Doctors differ, patients die 

message: 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)
thread_title:  Nonhomogeneous equation with constant coefficients 

message: 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)
thread_title:  Very minor edit 

message: In particular, the following solutions can be constructed
: y_{0'} = \tfrac{1}{2} \left (A_0 e^{i k x} + A_1 e^{-i k x} \right ) = C_0 \cos \left (\tfrac{kx}{2i} \right ) = C_1 \sin (k x). 
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 y_{0'} 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.  (y_{1'}?  y_{0}?)
I don't think the cosine form should have a denominator of 2i in the argument of \cos.
Jmichael ll (talk) 02:37, 22 May 2013 (UTC)
replies: text: y_{0'} = {C_0 e^{i k x} + C_0 e^{-i k x} \over 2} = C_0 \cos (k x) 
--Salix (talk): 04:28, 22 May 2013 (UTC)
thread_title:  Harmonic Oscillator equivalent solutions mangled 

message: 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. 
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)
thread_title:  First Order Equations with Varying Coefficients, Example, alternative equation 

message: 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)
thread_title:  Introductory Paragraph Misleading 

message: The first paragraph of the "Introduction" reads
Linear differential equations are of the form
 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).
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)
thread_title:  "Of the Same Nature" phrase is unclear 

message: 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)
replies: text: Hi  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? Wandalen (talk) 13:39, 3 June 2017 (UTC)
thread_title:  Obstruction to pending merge 

message: 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)
replies: text: It could be a good article, but on Wikibooks or Wikiversity.  — Arthur Rubin  (talk) 21:13, 8 June 2017 (UTC)
thread_title:  My 5 cents 

message: I have rewritten the whole article in the spirit of the new lead. The objective was
* Avoiding confusing change of notation (passing many time from operator language, with several notations to equation language. 
* Introducing every concept with as few technicalities as possible (in particular starting with explicit equations)
* Removing examples that do not provide insight, or changing them for reflecting the section where there are displayed
* Explaineing how the different methods are related.
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)
thread_title:  Rewrite of the article 

message: 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)
thread_title:  Rating of the article 

message: 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)
replies: text: 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 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. D.Lazard (talk) 20:40, 20 July 2018 (UTC)

text: 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 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.)  —Toby Bartels (talk) 23:35, 27 August 2018 (UTC)
thread_title:  Continuous coefficients? 

message: 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:
* William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th ed., Wiley, 2012.
:* "characteristic equation" is used repeatedly for differential equations on pp. 140-143, 145, 158, 160-163, 228, etc.
:* "characteristic equation" is used repeatedly for matrices on pp. 384, 385, 408, etc.
:* the link between the "characteristic equation" used for differential equations and the "characteristic equation" used for matrices is described on pp. 406-408.
:* "characteristic polynomial" is used repeatedly for differential equations on pp. 186, 228, 230, 236, 237, 240, etc.
* Martin Braun, Differential Equations and Their Applications, 4th ed., Springer, 1993.
:* "characteristic equation" is used repeatedly for differential equations on pp. 138, 139, 141, 145, 148, etc.
:* "characteristic equation" is used for matrices only once on pp. 351.
:* "characteristic polynomial" is used repeatedly for matrices on pp. 334, 338, 339, 341, 342, 345, etc.
* Richard Courant and Fritz John, Introduction to Calculus and Analysis, Vol. II/2, Chapter 6 "Differential Equations", Springer, 2000.
:*neither "characteristic equation" nor "characteristic polynomial" are used.
* Virginia Noonburg, Differential Equations: From Calculus to Dynamical Systems, 2nd ed., Vol. 43, MAA Press, 2019.
:* "characteristic polynomial" is used for differential equations on pp. 88, 91, 126.
:* "characteristic polynomial" is used for matrices on pp. 160, 388.
* Shepley L. Ross, Introduction to Ordinary Differential Equations, 4th ed., Wiley, 1989.
:* "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.
:* "characteristic equation" is used repeatedly for matrices on pp. 320-327, 383, 385, 386, etc.
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:
# (i) We leave this article as it is; (ii) we create a disambiguation page "Characteristic polynomial" (in the same vein as the disambiguation page Characteristic equation); (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.
# (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.
# (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 Characteristic equation); (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.
# We do nothing and let students in confusion.
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)
replies: text: Here as well as in the matrix case, a "caracteristic equation" and a "characteristic polynomial" differ only by having or not "" 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. D.Lazard (talk) 12:03, 13 July 2020 (UTC)
thread_title:  Characteristic Equation, Characteristic Polynomial 

message: Currently the section "First-order equation with variable coefficients" concludes
:"\frac{d}{dx}\left(ye^{-F}\right)= ge^{-F}.
Thus, the general solution is
:y=ce^F + e^F\int ge^{-F}dx,
where  is a constant of integration, and ."
It seems to me that c is already contained in the integral. If we say that H is an antiderivative of ge^{-F}, we get
: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)
And then the constant simply gets absorbed into the integral.__Gamren (talk) 11:01, 12 February 2021 (UTC)
replies: text: In this context, an indefinite integral must be interpreted as any 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. D.Lazard (talk) 11:43, 12 February 2021 (UTC)
thread_title:  Redundant integration constant?