Talk:Linear differential equation

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

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

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 ${\displaystyle d/dt}$ 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

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

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

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

and then with a little rewriting we have

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

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

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 ${\displaystyle D^{1}}$ and ${\displaystyle D^{0}}$ 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 ${\displaystyle e^{zx}}$, for possibly-complex values of ${\displaystyle z}$. 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

${\displaystyle {\frac {d^{n}y}{dx^{n}}}+A_{1}{\frac {d^{n-1}y}{dx^{n-1}}}+\cdots +A_{n}y=0}$

we set ${\displaystyle y=e^{zx}}$, leading to

${\displaystyle z^{n}e^{zx}+A_{1}z^{n-1}e^{zx}+\cdots +A_{n}e^{zx}=0.}$

Specifically, for consistency with the introductory text it is more appropriate to use the exponential ${\displaystyle e^{rt}}$ 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

${\displaystyle {\frac {d^{N}y}{dt^{N}}}+A_{1}{\frac {d^{N-1}y}{dt^{N-1}}}+\cdots +A_{N}y=0}$

we set ${\displaystyle y=e^{rt}}$, leading to

${\displaystyle r^{N}e^{rt}+A_{1}r^{N-1}e^{rt}+\cdots +A_{N}e^{rt}=0.}$

and this factors as

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

Since ${\displaystyle e^{rt}}$ can not be zero then we have the classic characteristic equation:

${\displaystyle r^{N}+A_{1}r^{N-1}+\cdots +A_{N}=0.}$

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 ${\displaystyle e^{zx}}$, for possibly-complex values of ${\displaystyle z}$. 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 ${\displaystyle z=x+iy}$? 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: ${\displaystyle y'=2x}$, one of it's solutions is ${\displaystyle y_{0}=x^{2}}$, but ${\displaystyle 2y_{0}}$ is not solution of original equation: ${\displaystyle (2x^{2})'=4x\neq 2x}$.

Harmonic Oscillator equivalent solutions mangled

In particular, the following solutions can be constructed

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

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 ${\displaystyle y_{0'}}$ 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. (${\displaystyle y_{1'}}$? ${\displaystyle y_{0''}}$?)

I don't think the cosine form should have a denominator of ${\displaystyle 2i}$ in the argument of ${\displaystyle \cos }$.

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,

${\displaystyle y_{0}=A_{0}e^{ikx}}$
and
${\displaystyle y_{1}=A_{1}e^{-ikx}.}$

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

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

and

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

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

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

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

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

--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 ${\displaystyle P_{0}(x)y^{(n)}+P_{1}(x)y^{(n-1)}+\cdots +P_{n-1}(x)y'+P_{n}(x)y=F(x)}$ [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. [1]

References

1. ^ Berberan-Santos, Mário N. (2010). "Green’s function method and the first-order linear differential equation". J Math Chem. 48 (2): 175–178. doi:10.1007/s10910-010-9678-2.

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 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)