Changing variables in integrals?

[Chen]

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 v²_a, v²_b and v²_c 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?

Thanks,
Chen

 

[lalbatros]

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)

 

[Chen]

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

 

[lalbatros]

It is just by chaining the maps!
By definition

y²(x) = (v²(x)-va²)/(vb²-va²)