Talk:Variation of parameters

message: 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)
thread_title:  Example 

message: 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)
replies: text: Any first-order linear ODE can be solved with little fuss — see Ordinary differential equation#General solution method for first-order linear ODEs — such that techniques like the method of variation of parameters are unnecessary. Ruakh 15:29, 1 February 2006 (UTC)
replies: text: It should be noted that variation of parameters does work for first order ODE just as it does for 2nd order and higher. But you are correct that conceptually easier techniques are available there.163.118.103.199 (talk) 18:34, 5 March 2012 (UTC)
thread_title:  Using this for first-order ODEs 

message: In the beginning of the Technique section, the article says u_1 and u_2 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)
replies: text: Good call. I've fixed the article now. Ruakh 13:30, 15 September 2006 (UTC)
replies: text: Thanks! Lavaka 01:55, 20 September 2006 (UTC)
thread_title:  u is homogenous solution? 

message: The first example calculation involves an integral containing f(x),
:A(x) = - \int {1\over W} u_2(x) f(x)\,dx,\; B(x) = \int {1 \over W} u_1(x)f(x)\,dx
Nowhere up to this point is f(x) defined; in fact (it refers to the right-hand-side of the ODE) the function on the right-hand-side has previously been called b(x).  —Preceding unsigned comment added by 68.49.223.78 (talk) 14:57, 8 March 2010 (UTC)
thread_title:  Mistake in the First Example 

message: 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)
thread_title:  Why? 

message: 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
:mx + \mu x + k x = b(t)
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 [t,t+dt] the effect of imparting an additional momentum b(t)dt 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)
thread_title:  Newton's law and variation of parameters 

message: 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)
thread_title:  Merge content from [[Duhamel's principle]]