# 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! $ax + by + c = 0$ 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

$a_n(x) D^n y(x) + a_{n-1}(x)D^{n-1} y(x) + \cdots + a_1(x) D y(x) + a_0(x) y(x) =f(x)$

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 $d/dt$ 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

$g = p_0 D^2 f(x) + p_1D^1 f(x) + p_2 D^0 f(x)\!$

and then with a little rewriting we have

$g = [p_0 D^2 + p_1D^1 + p_2 D^0]f(x) = L[f(x)]\!$

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 $\ x^2 y'' + x y' + (x^2 - n^2) y = 0,\ n \in \mathbb{R}$.
• 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)

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 $e^{z x}$, 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)

### 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 $e^{z x}$, 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)

## 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: $y' = 2 x$, one of it's solutions is $y_0 = x^2$, but $2 y_0$ is not solution of original equation: $(2 x^2)' = 4 x \ne 2 x$.

## Harmonic Oscillator equivalent solutions mangled

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)

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

The solutions are, respectively,

$y_0 = A_0 e^{i k x}$
and
$y_1 = A_1 e^{-i k x}.$

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

$y_{0'} = {A_0 e^{i k x} + A_1 e^{-i k x} \over 2} = C_0 \cos (k x)$

and

$y_{1'} = {A_0 e^{i k x} - A_1 e^{-i k x} \over 2 i} = C_1 \sin (k x).$

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

$y_H = C_0 \cos (k x) + C_1 \sin (k x).$

which is still not perfect as we need $A_0=A_1=C_0$ for equality to hold. It may be better to write

$y_{0'} = {C_0 e^{i k x} + C_0 e^{-i k x} \over 2} = C_0 \cos (k x)$

--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)

## 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]

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.