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Solve a differential equation

  1. Dec 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve this differential equation

    http://www.texify.com/img/%5CLARGE%5C%21%5Cfrac%7B2%7D%7Bx%5E2%7D%28%5Cfrac%7Bdx%7D%7Bds%7D%29%5E2-%5Cfrac%7B1%7D%7Bx%7D%20%5Cfrac%7B%7Bd%5E2%7Dx%7D%7Bd%7Bs%5E2%7D%7D%3D1.gif [Broken]


    (thanks Mark)


    where x is unknown, s represents the metric http://www.texify.com/img/%5CLARGE%5C%21ds%5E2%3D%281/x%5E2%29dx%5E2%2B%281/x%5E2%29dy%5E2.gif [Broken]

    2. Relevant equations
    just that equation


    3. The attempt at a solution

    I tried writing out (d^2/ds^2)(1/x), but it wasn't right. x is unknown, s represents the metric ds^2=(1/x^2)dx^2+(1/x^2)dy^2

    This might work if we use a substitution, I just don't know what.
    Thanks
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Dec 18, 2009 #2

    Mark44

    Staff: Mentor

    Added tex tags.
     
    Last edited by a moderator: May 4, 2017
  4. Dec 19, 2009 #3
    Figured it out.

    Use
    http://www.texify.com/img/%5CLARGE%5C%21%5Cfrac%7B%7Bd%5E2%7Du%28x%29%7D%7Bd%7Bs%5E2%7D%7D%3D%5Cfrac%7Bd%7D%7Bds%7D%20%28%5Cfrac%7Bdu%7D%7Bdx%7D%20%5Cfrac%7Bdx%7D%7Bds%7D%29.gif [Broken]
    Oops, I guess I don't know how to paste in the text.

    write out: d^2(u(x))/ds^2
    then if we let u=(1/x) then plugging in we find that (d^2(1/x)/ds^2)=1/x

    that's all I needed (solving for x)
     
    Last edited by a moderator: May 4, 2017
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