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Solving a differential to show the homogeneity of space.

  1. Feb 2, 2016 #1
    1. The problem statement, all variables and given/known data
    The final part of the problem I am trying to solve requires the proof of the following equation:
    [tex]\frac{d}{dr}(\frac{rf'(r)-f(r)+f^2(r)}{r^2 f^2(r)})=0[/tex]



    2. Relevant equations
    I've been given the ansatz:
    [tex]f(r)=(1-kr^2)^{-1}[/tex]
    leading to
    [tex]f'(r)=2krf^2(r)[/tex]
    [tex]f''(r)=2kf^2(r)+8(kr)^2f^3(r)[/tex]
    3. The attempt at a solution
    Using the quotient rule on the first equation and cancelling some terms and the denominator gets me to:
    [tex](rf)^2(rf''+2ff')-(rf'-f+f^2)(2rf^2+2r^2ff')=0[/tex]
    Expanding, cancelling and subbing in for f' & f'' leads me to:
    [tex]2kr^3f^4+8k^2r^5f^5+2rf^3-2rf^4-4kr^4f^3=0[/tex]

    I have tried subbing in the ansatz value for f, but I still can't get the terms to cancel out. Would anybody be able to point out where I've made a mistake/ what I've missed?
     
  2. jcsd
  3. Feb 2, 2016 #2

    RUber

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    If you were to start with your ansatz, then the problem looks like:
    ## \frac{d}{dr}\left( \frac{ r(2kr)f^2 - f + f^2}{r^2 f^2 }\right) =0 ##
    Which could just as well be written as:
    ##\left( \frac{ r(2kr)f^2 - f + f^2}{r^2 f^2 }\right) =C ##
    Separate the fractions and you have: *edited to correct sign error*
    ##\left( \frac{ r(2kr)f^2 }{r^2 f^2 } + \frac{ -f + f^2}{r^2 f^2 }\right) =C ##
    That first fraction is clearly a constant...do some algebra on the second...and I think you might find your solution.
     
    Last edited: Feb 2, 2016
  4. Feb 2, 2016 #3
    Ah thanks! I've got there now. On the solution sheet they seem to suggest a different method using f''(r) but don't show it explicitly but this definitely seems valid. (Though I think the + should be a - in your third equation after splitting the fraction.)
     
  5. Feb 2, 2016 #4

    RUber

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    I agree on both points. I was hasty in splitting the fractions and it did sound like the problem was enticing you to take the complicated derivative.
     
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