Talk:Linear differential equation

Moved material from ordinary differential equations

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)

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

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 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)

Too Technical

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! $ax+by+c=0$ is a good starting position for a line in 2D. And, $ax+by=0$ 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

$a_{n}(x)D^{n}y(x)+a_{n-1}(x)D^{n-1}y(x)+\cdots +a_{1}(x)Dy(x)+a_{0}(x)y(x)=f(x)$

where D is the differential operator d/dx (i.e. Dy = y' , D²y = y",... ), and the a_i 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 a_n 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 $d/dt$ 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

$g=p_{0}f''+p_{1}f'+p_{2}f\!$

to explain the linear transformation of the function f into g. So let us compare with

$g=p_{0}D^{2}f(x)+p_{1}D^{1}f(x)+p_{2}D^{0}f(x)\!$

and then with a little rewriting we have

$g=[p_{0}D^{2}+p_{1}D^{1}+p_{2}D^{0}]f(x)=L[f(x)]\!$

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)

Classical examples

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 $\ x^{2}y''+xy'+(x^{2}-n^{2})y=0,\ n\in \mathbb {R}$.
  • Bessel function  : $x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0$
  • ==>Linear operator form: $\left[x^{2}D^{2}+xD^{1}+(x^{2}-\alpha ^{2})D^{0}\right]y=0,\,\!$

• 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 :$x^{2}{\frac {d^{2}y}{dx^{2}}}+ax{\frac {dy}{dx}}+by=0\,$
  • ==> Linear operator form: $\left[x^{2}D^{2}+axD^{1}+bD^{0}\right]y=0,\,\!$

• Hermitesche Differentialgleichung $\ y''-2xy'+2ny=0,\ n\in \mathbb {Z}$.
  • Hermitian polynomials :$L[u]=u''-xu'=-\lambda u$
  • ==> Linear operator form (from the equation on German site): $\left[D^{2}-2xD^{1}+2nD^{0}\right]\,y=0\!$
  • 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 :$z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-abw=0.$
  • ==> Linear operator form(English version) : $\left[z(1-z)D^{2}+\left(c-(a+b+1)z\right)D^{1}-abD^{0}\right]w=0.$

• Laguerresche Differentialgleichung $x\,y''+(1-x)\,y'+ny=0,\ n\in \mathbb {N} _{0}$.
  • 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) : $\left[(1-x^{2})D^{2}-2xD^{1}+n(n+1)D^{0}\right]\,y=0$

• Tschebyschowsche Differentialgleichung $\ (1-x^{2})y''-xy'+n^{2}y=0$.
  • Chebyshev polynomials: $(1-x^{2})\,y''-x\,y'+n^{2}\,y=0\,\!$
  • ==> Linear operator form : $\left[\,(1-x^{2})D^{2}-xD^{1}+n^{2}D^{0}\right]\,y=0$
  • and: $(1-x^{2})\,y''-3x\,y'+n(n+2)\,y=0\,\!$
  • ==> Linear operator form : $\left[\,(1-x^{2})D^{2}-3xD^{1}+n(n+2)D^{0}\right]\,y=0$

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)

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 $D^{1}$ and $D^{0}$ 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)

References

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:

1. Gershenfeld, Neil (1999), The Nature of Mathematical Modeling, Cambridge, UK.: Cambridge University Press, ISBN 978-0521-570954

Then I will add in other reference works as appropriate. — Михал Орела (talk) 13:12, 14 September 2009 (UTC)

Homogeneous equations with constant coefficients

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 $e^{zx}$, 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 $y=e^{zx}$, 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 $e^{rt}$ 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 $y=e^{rt}$, 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 $e^{rt}$ 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)

Doctors differ, patients die

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 $e^{zx}$, 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)

Nonhomogeneous equation with constant coefficients

[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)

Very minor edit

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)

vector space?

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)

I agree with Sandrobt - it should be fixed. Example is: $y'=2x$, one of it's solutions is $y_{0}=x^{2}$, but $2y_{0}$ is not solution of original equation: $(2x^{2})'=4x\neq 2x$.

Harmonic Oscillator equivalent solutions mangled

In particular, the following solutions can be constructed

$y_{0'}={\tfrac {1}{2}}\left(A_{0}e^{ikx}+A_{1}e^{-ikx}\right)=C_{0}\cos \left({\tfrac {kx}{2i}}\right)=C_{1}\sin(kx).$

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)

Well spotted it was introduced in this edit in February [2]. I've reverted it. The current text is

The solutions are, respectively,

$y_{0}=A_{0}e^{ikx}$

and

$y_{1}=A_{1}e^{-ikx}.$

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

$y_{0'}={A_{0}e^{ikx}+A_{1}e^{-ikx} \over 2}=C_{0}\cos(kx)$

and

$y_{1'}={A_{0}e^{ikx}-A_{1}e^{-ikx} \over 2i}=C_{1}\sin(kx).$

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

$y_{H}=C_{0}\cos(kx)+C_{1}\sin(kx).$

which is still not perfect as we need $A_{0}=A_{1}=C_{0}$ for equality to hold. It may be better to write

$y_{0'}={C_{0}e^{ikx}+C_{0}e^{-ikx} \over 2}=C_{0}\cos(kx)$

--Salix (talk): 04:28, 22 May 2013 (UTC)

Linear differential equations include inhomogeneous linear differential equations

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)

Agreed. In support here are two quotations from C. H. Edwards and D. E. Penny, Elementary Differential Equations, 4ed, Prentice-Hall, 2000, ISBN 0-13-011290-9:

• ... 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 ... [page 94]
• ... the general nth-order linear differential equation of the form $P_{0}(x)y^{(n)}+P_{1}(x)y^{(n-1)}+\cdots +P_{n-1}(x)y'+P_{n}(x)y=F(x)$ [page 109]

As Sandrobt points out in the "vector space?" section above, it is not true that "linear differential equations [have] 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 part of the article that deals with nonhomogeneous equations. JonH (talk) 16:48, 24 March 2015 (UTC)

First Order Equations with Varying Coefficients, Example, alternative equation

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)

Introductory Paragraph Misleading

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)

"Of the Same Nature" phrase is unclear

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)

Exponential response formula

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)

Discussion

• Support Merge. Basically, there's less here than meets the eye. The current state of the target article contains more 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 Wikipedia. ? Arthur Rubin (talk) 05:39, 30 May 2017 (UTC)
• strong Support Merge for not simply honoring coining acronyms as justification for creating new WP articles on formulae, arising in solving special cases of certain classes of DEQ. I also agree to the arguments above. Purgy (talk) 06:56, 30 May 2017 (UTC)
• Don't 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 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 Wikipedia society is more tolerant to newcomers than you two are. Wandalen (talk) 19:15, 30 May 2017 (UTC)

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. Purgy (talk) 06:39, 31 May 2017 (UTC)

• Mr. Rubin and Mr. Prugy, I vote against 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.

ERF was mentioned in Wikipedia cite, titled Linear differential equation. 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.

"Can not you use the ERF method if P (r) is 0? or what is it if we can use?

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 on the Wikipedia.

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

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 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 Wikipedia.

Finally, let's discuss the merge problem.

Linear differential equation 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.

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

Even if it does, it will bring boredom of readers. Therefore, each parts are described in depth by its own cites. For example, Method of undetermined coefficients and Variation of parameters. If the merging problem is constantly discussed, I think that both of above cites should be merged into a linear differential equation cite. However, there are exist two cites independently.

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

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

Best regards.

Y.phillip (talk) 2:38, 6 June 2017 (UTC)

• I am against, per my explanation on the talk page discussion, mainly due to the face that I believe long articles 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. --EngiZe (talk) 07:55, 24 June 2017 (UTC)

Obstruction to pending merge

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)

Hi @Purgy Purgatorio: 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)

My 5 cents

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)

It could be a good article, but on Wikibooks or Wikiversity. — Arthur Rubin (talk) 21:13, 8 June 2017 (UTC)

Major rewrite of lead

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)

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 it:

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

- 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.

- Referring to an "independent variable" is unnecessary and distracting, and carries with it a heap of outdated misunderstandings.

- I claim that at this here level the term "value", when addressing a full blown "map", is disturbing.

- 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.

- Raising to powers smaller than one is de rigeur?

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

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

---

Linear differential equations are differential equations which equate the result of applying a linear differential operator to some unknown function with some other function, called source term or response. 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 $(y_{i})_{i\in I}$ for some set $I$ of parameters that satisfies the differential equation.

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 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 $L$, the unknown function with $y$ and the response with $f$, the LDEQ can compactly be written as

$Ly=f.$

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 $f=0$

$Ly=0$

is of great importance. The solutions to (homogeneous) linear differential equations form a vector space (unlike non-linear differential equations).

Linear differential equations can be ordinary (ODEs) or partial (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 constant coefficients which allow for closed solvability in the homogeneous case.

---

End of my reply. Purgy (talk) 09:51, 27 January 2018 (UTC)

I’ve gone to WT:MATH and requested participation in this thread. Loraof (talk) 17:20, 27 January 2018 (UTC)

Previous lead was a mess, and was difficult to understand even for experts. Purgy version is better, but much too technical. 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. D.Lazard (talk) 18:14, 27 January 2018 (UTC)

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) depends on x and so it is not constant. Perhaps it could say "if the function b(x) is zero". JonH (talk) 21:03, 27 January 2018 (UTC)

Thoughts on the new lead

- 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)

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.

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

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

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 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.

By the way, this article must also have a section about holonomic functions, 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 Dynamic Dictionary of Mathematical Functions) D.Lazard (talk) 12:05, 28 January 2018 (UTC)

Rewrite of the article

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)

Rating of the article

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)

Continuous coefficients?

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)

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)

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)

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)

Characteristic Equation, Characteristic Polynomial

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:

1. (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.
2. (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.
3. (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.
4. 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)

Here as well as in the matrix case, a "caracteristic equation" and a "characteristic polynomial" differ only by having or not "= 0" 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)

Redundant integration constant?

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 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 $ge^{-F}$, 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)

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)

Proposed merge

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.

Closed as withdrawn by the nominator. I have PROD'ded Armour formula instead since there is clearly no material there worth merging. Felix QW (talk) 21:30, 13 January 2022 (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)

A merge could made with Integrating factor#Solving first order linear ordinary differential equations. And this article should mention the term "integrating factor" with a link to its article. JonH (talk) 21:32, 6 January 2022 (UTC)

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

Armour formula, Examples of differential equations#Non-separable (non-homogeneous) first-order linear ordinary differential equations, Linear differential equation#First-order equation with variable coefficients and Integrating factor#Solving first order linear ordinary differential equations. 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 withdrawing my original merge proposal since there is nothing to merge. Felix QW (talk) 20:26, 13 January 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.