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Differential equation

  1. Mar 31, 2009 #1
    1. The problem statement, all variables and given/known data

    How to solve the following DE:
    [tex]\frac{1}{\sqrt{1+(dy/dx)^{2}}}=\frac{2y^{2}}{2}+C[/tex]?
     
  2. jcsd
  3. Mar 31, 2009 #2

    CompuChip

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    I suppose solving it for dy/dx might enable you to do a separation of variables...

    I.e. (since you are posting this in advanced physics): write
    dy/dx = f(y)
    for some function f only depending on y; then integrate
    dx = dy / f(y)
    and invert to find y(x).

    Granted, it's probably easier said than done, but you can give it a try.
     
  4. Apr 2, 2009 #3
    It is indeed separable. I get it into the following form, but don't know how to integrate
    [tex]dx=\sqrt{\frac{((2/\gamma)y^{2}+C)^{2}}{1-((2/\gamma)y^{2}+C)^{2}}}dy[/tex]
     
  5. Apr 2, 2009 #4
    This is an elegant problem.

    Superb.

    First: Lets try to make the equation a bit less horrendous.

    Take [tex]\sqrt{1-((2/\gamma)y^{2}+C)^{2}} = t [/tex]

    Proceed with that. Simplify it well and then take

    [tex] t= sin\theta[/tex]

    Simplify it and then use De moivre's theorem.
     
  6. Apr 2, 2009 #5
    May i know the name of the book.
     
  7. Apr 13, 2009 #6
    But the problem with this substitution is that there is a second power of y in the square root. Thus there will be a term including y for the expression for dt...
     
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