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In the first example, it says "So, we obtain u1=e-2x, and u2=xe-2x." This is in my text too, but I never understood why. Could someone add a short comment saying why (e.g. "So, we obtain u1=e^-2x (see blahs_rule) and we introduce an x term (see blah2s_rule) as well to yield u2=xe^-2x"). 75.128.252.106 ( talk) 03:50, 8 February 2008 (UTC)
Can't this also be used for 1st order ODE's? Perhaps I'm thinking of a different method, but I was just looking at my Differential Equations Text, and the method seems the same, except for first order equations. Any thoughts? Gershwinrb 06:34, 1 February 2006 (UTC)
Should we also have examples for systems of equations and/or higher order ODEs? jleto
In the beginning of the Technique section, the article says and are "solutions" to the equation. It really means solutions to the homogenous equation, right? If not, I'm totally confused. This should be changed and made clear. Lavaka 05:32, 15 September 2006 (UTC)
I deleted the following text from the Ordinary differential equation where it consumed far too much space. I copied it here in case anyone is able to salvage some parts and integrate them into this article. MathMartin 20:24, 11 December 2006 (UTC)
As explained above, the general solution to a non-homogeneous, linear differential equation can be expressed as the sum of the general solution to the corresponding homogenous, linear differential equation and any one solution to .
Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to , having already found the general solution to . Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use.
For a second-order equation, the method of variation of parameters makes use of the following fact:
Let p(x), q(x), and g(x) be functions, and let and be solutions to the homogeneous, linear differential equation . Further, let u(x) and v(x) be functions such that and for all x, and define . Then is a solution to the non-homogeneous, linear differential equation .
To solve the second-order, non-homogeneous, linear differential equation using the method of variation of parameters, use the following steps:
The method of variation of parameters can also be used with higher-order equations. For example, if , , and are linearly independent solutions to , then there exist functions u(x), v(x), and w(x) such that , , and . Having found such functions (by solving algebraically for u'(x), v'(x), and w'(x), then integrating each), we have , one solution to the equation .
Solve the previous example, Recall . From technique learned from 3.1, LHS has root of that yield , (so , ) and its derivatives
where the Wronskian
were computed in order to seek solution to its derivatives.
Upon integration,
Computing and :
In the fith equation of the section Method_of_variation_of_parameters#Method_of_variation_of_parameters it should be "b(x)" rather than "-b(x)". But i am not completely sure. —The preceding unsigned comment was added by 141.35.186.111 ( talk • contribs) 13:33, 2 February 2007 (UTC).
I think the integrals should be changed to use dummy variables -- as written now, they are misleading. The current format is but I'd much rather see a dummy variable, e.g. or at least Anyone interested in redoing this? -- Lavaka 18:20, 17 April 2007 (UTC)
The first example calculation involves an integral containing ,
Nowhere up to this point is defined; in fact (it refers to the right-hand-side of the ODE) the function on the right-hand-side has previously been called . —Preceding unsigned comment added by 68.49.223.78 ( talk) 14:57, 8 March 2010 (UTC)
I am not sure, whether the statement in parenthesis is 100% correct.
(results from substitution of (iii) into the homogeneous case (ii); )
How is one supposed to conclude this from the described substitution?? —Preceding unsigned comment added by 93.104.136.99 ( talk) 14:47, 19 September 2010 (UTC)
why equation 3 is true? Please tell, if you know, and have some time. eq 2 looks like definition of null space, and the y sub i forms a basis of the null space, a subspace of the vector space of — Preceding unsigned comment added by 108.236.198.181 ( talk) 23:34, 2 June 2013 (UTC)
It seems to me that someone should write a brief motivation (for second order ODEs) of the method via Newton's law. The left-hand side of a nonhomogeneous second-order ODE
is the net force on a mass attached to a damped spring. The right-hand side is the external force applied to the spring. The method of variation of parameters is nothing more than cumulatively adding to the solution during each time interval the effect of imparting an additional momentum to the mass. It would be nice if someone could track down a reference for this (and hopefully a clearer explanation) and add it to the article. The same essential idea applies more generally to Duhamel's principle. Sławomir Biały ( talk) 18:42, 24 June 2013 (UTC)
Also the matrix formulation of variation of parameters is conspicuously absent from the article. Sławomir Biały ( talk) 18:45, 24 June 2013 (UTC)
All the references I could find that mentioned both variation of parameters and Duhamel's principle said they were equivalent. Even the variation of parameters article says that they're related. I tried tracking down the source of "Duhamel's principle" and they just mention the superposition principle (again something done in variation of parameters). Variation of parameters seems like the more common term in overall usage. -- Mathnerd314159 ( talk) 05:50, 7 April 2017 (UTC)
No; in ODEs, Variation of Parameters refers to a specific method that finds the coefficients of the particular solution to a differential equation. This is not the same thing as Duhamel's Principle at all. --Review day in Math 240 — Preceding
unsigned comment added by
216.125.152.241 (
talk)
19:42, 7 November 2017 (UTC)
Hello,
I've studied mathematics through my life and I find it hard to remember something which is true but feels like the answer came from the sky. Most of the time great mathematical results can be proved and summed up in three lines, but then no trace is left of the days/weeks/months/years/letter exchanges between mathematicians which led up to those brilliant short truths. I feel variation of parameters is one of these things where the proof is relatively easy to understand, but it looks like a recipe someone pulled out of his/her hat. I appreciate the history section of the artice about the study of planets, but it sheds little light (for me) on how someone thought of trying to replace a constant with a function. Did someone try it by accident ? Does it have something to do with "perturbated orbits"? I understand it can be hard to sum up in an article a long history of research and discovery, but I just hoped I could find somewhere more explanations about how and why somebody came up with this idea. Anyway, I'm sorry if I come off as complaining and am not offering solutions here, but if anyone has a link which could enlighten me on how someone thought of this method, I'd be very happy to click on it ! Thank you very much.-- ByteMe666 ( talk) 19:45, 17 May 2018 (UTC)
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
In the first example, it says "So, we obtain u1=e-2x, and u2=xe-2x." This is in my text too, but I never understood why. Could someone add a short comment saying why (e.g. "So, we obtain u1=e^-2x (see blahs_rule) and we introduce an x term (see blah2s_rule) as well to yield u2=xe^-2x"). 75.128.252.106 ( talk) 03:50, 8 February 2008 (UTC)
Can't this also be used for 1st order ODE's? Perhaps I'm thinking of a different method, but I was just looking at my Differential Equations Text, and the method seems the same, except for first order equations. Any thoughts? Gershwinrb 06:34, 1 February 2006 (UTC)
Should we also have examples for systems of equations and/or higher order ODEs? jleto
In the beginning of the Technique section, the article says and are "solutions" to the equation. It really means solutions to the homogenous equation, right? If not, I'm totally confused. This should be changed and made clear. Lavaka 05:32, 15 September 2006 (UTC)
I deleted the following text from the Ordinary differential equation where it consumed far too much space. I copied it here in case anyone is able to salvage some parts and integrate them into this article. MathMartin 20:24, 11 December 2006 (UTC)
As explained above, the general solution to a non-homogeneous, linear differential equation can be expressed as the sum of the general solution to the corresponding homogenous, linear differential equation and any one solution to .
Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to , having already found the general solution to . Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use.
For a second-order equation, the method of variation of parameters makes use of the following fact:
Let p(x), q(x), and g(x) be functions, and let and be solutions to the homogeneous, linear differential equation . Further, let u(x) and v(x) be functions such that and for all x, and define . Then is a solution to the non-homogeneous, linear differential equation .
To solve the second-order, non-homogeneous, linear differential equation using the method of variation of parameters, use the following steps:
The method of variation of parameters can also be used with higher-order equations. For example, if , , and are linearly independent solutions to , then there exist functions u(x), v(x), and w(x) such that , , and . Having found such functions (by solving algebraically for u'(x), v'(x), and w'(x), then integrating each), we have , one solution to the equation .
Solve the previous example, Recall . From technique learned from 3.1, LHS has root of that yield , (so , ) and its derivatives
where the Wronskian
were computed in order to seek solution to its derivatives.
Upon integration,
Computing and :
In the fith equation of the section Method_of_variation_of_parameters#Method_of_variation_of_parameters it should be "b(x)" rather than "-b(x)". But i am not completely sure. —The preceding unsigned comment was added by 141.35.186.111 ( talk • contribs) 13:33, 2 February 2007 (UTC).
I think the integrals should be changed to use dummy variables -- as written now, they are misleading. The current format is but I'd much rather see a dummy variable, e.g. or at least Anyone interested in redoing this? -- Lavaka 18:20, 17 April 2007 (UTC)
The first example calculation involves an integral containing ,
Nowhere up to this point is defined; in fact (it refers to the right-hand-side of the ODE) the function on the right-hand-side has previously been called . —Preceding unsigned comment added by 68.49.223.78 ( talk) 14:57, 8 March 2010 (UTC)
I am not sure, whether the statement in parenthesis is 100% correct.
(results from substitution of (iii) into the homogeneous case (ii); )
How is one supposed to conclude this from the described substitution?? —Preceding unsigned comment added by 93.104.136.99 ( talk) 14:47, 19 September 2010 (UTC)
why equation 3 is true? Please tell, if you know, and have some time. eq 2 looks like definition of null space, and the y sub i forms a basis of the null space, a subspace of the vector space of — Preceding unsigned comment added by 108.236.198.181 ( talk) 23:34, 2 June 2013 (UTC)
It seems to me that someone should write a brief motivation (for second order ODEs) of the method via Newton's law. The left-hand side of a nonhomogeneous second-order ODE
is the net force on a mass attached to a damped spring. The right-hand side is the external force applied to the spring. The method of variation of parameters is nothing more than cumulatively adding to the solution during each time interval the effect of imparting an additional momentum to the mass. It would be nice if someone could track down a reference for this (and hopefully a clearer explanation) and add it to the article. The same essential idea applies more generally to Duhamel's principle. Sławomir Biały ( talk) 18:42, 24 June 2013 (UTC)
Also the matrix formulation of variation of parameters is conspicuously absent from the article. Sławomir Biały ( talk) 18:45, 24 June 2013 (UTC)
All the references I could find that mentioned both variation of parameters and Duhamel's principle said they were equivalent. Even the variation of parameters article says that they're related. I tried tracking down the source of "Duhamel's principle" and they just mention the superposition principle (again something done in variation of parameters). Variation of parameters seems like the more common term in overall usage. -- Mathnerd314159 ( talk) 05:50, 7 April 2017 (UTC)
No; in ODEs, Variation of Parameters refers to a specific method that finds the coefficients of the particular solution to a differential equation. This is not the same thing as Duhamel's Principle at all. --Review day in Math 240 — Preceding
unsigned comment added by
216.125.152.241 (
talk)
19:42, 7 November 2017 (UTC)
Hello,
I've studied mathematics through my life and I find it hard to remember something which is true but feels like the answer came from the sky. Most of the time great mathematical results can be proved and summed up in three lines, but then no trace is left of the days/weeks/months/years/letter exchanges between mathematicians which led up to those brilliant short truths. I feel variation of parameters is one of these things where the proof is relatively easy to understand, but it looks like a recipe someone pulled out of his/her hat. I appreciate the history section of the artice about the study of planets, but it sheds little light (for me) on how someone thought of trying to replace a constant with a function. Did someone try it by accident ? Does it have something to do with "perturbated orbits"? I understand it can be hard to sum up in an article a long history of research and discovery, but I just hoped I could find somewhere more explanations about how and why somebody came up with this idea. Anyway, I'm sorry if I come off as complaining and am not offering solutions here, but if anyone has a link which could enlighten me on how someone thought of this method, I'd be very happy to click on it ! Thank you very much.-- ByteMe666 ( talk) 19:45, 17 May 2018 (UTC)