Changing variables in integrals?

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Discussion Overview

The discussion revolves around the transformation of variables in integrals, specifically in the context of a paper involving an ordinary differential equation (ODE) and its integration. Participants are exploring the transition from one function to another and the implications for the limits of integration.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Chen expresses confusion regarding the transition from the original function v2 to the new function y2 and how this affects the limits of integration in equation (5.14).
  • One participant suggests a method for transforming the integral by substituting variables and simplifying the expression, indicating that the process is straightforward once the correct approach is known.
  • Chen acknowledges the assistance but remains uncertain about how the limits of integration change from v2(0) and v2(xi) to y(0) and y(xi), questioning the fundamental understanding of variable changes in integrals.
  • Another participant provides a definition for y²(x) in terms of v²(x), suggesting that the transformation of limits is a result of this relationship.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the transformation of limits in the integral, as Chen continues to seek clarification on this aspect.

Contextual Notes

There may be missing assumptions regarding the properties of the functions involved and the specific conditions under which the variable transformations are applied. The discussion does not resolve these uncertainties.

Who May Find This Useful

Readers interested in mathematical transformations in integrals, particularly in the context of differential equations and variable substitutions, may find this discussion relevant.

Chen
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Hi,

I'm reading the following paper:
http://prola.aps.org/abstract/PR/v127/i6/p1918_1
(It's physics, you don't really have to click that)

Somewhere along the way the authors make a transition that I can't understand. Basically they have an ODE (5.9), which is integrated to give equation (5.10). Then they define a new function, y2, to replace the original function, v2, and find a new integral equation for it (5.14). I'm bringing two screen shots from the article that show this transition. The only other thing you need to know is that v2a, v2b and v2c are the roots of the expression insider the root in the integral of (5.10).

Screen shots: (in order)
http://img59.imageshack.us/img59/1887/partajj6.png
http://img62.imageshack.us/img62/5359/partbyc9.png

Now, I tried following this transition myself and couldn't make any sense of it. Most of all, I don't understand how the final integral equation (5.14) has y in the limits of integration, and not y2 or something like that.

Help, please? :cry:

Thanks,
Chen
 
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Chen, this is straightforward.
Just write v²(1-v²)²-G² = (v²-va²)(v²-vb²)(v²-vc²) = (vb²-va²)y²(v²-va²+va²-vb²),(v²-va²+va²-vc²),
then continue replacing (v²-va²) = y² (vb²-va²) and express everything from y²,
then substitute this is the square root in the denominator,
then don't forget to go from d(v²) to dy (this will do a small simplification,
and its finished.

(unfortunately, I have no access to the original paper)
 
lalbatros,

Thanks for the help! You're right, it's pretty straightforward once you know the trick... so I was able to transform the root into the required form, and all the constants also turned out okay - but I'm still bothered by the limits of integration.
Maybe I don't understand something fundamental about this kind of operations, but how do I end up with y(0) and y(xi) in the limits? Starting with v2(0) and v2(xi), can you please explain how these limits transform?
(It's not like I don't know how to change variables inside integrals, but for some reason this seems weird to me...)

Thank you! :-)
Chen
 
It is just by chaining the maps!
By definition

y²(x) = (v²(x)-va²)/(vb²-va²)
 

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