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A second order nonlinear ordinary differential equation

  1. Mar 18, 2010 #1
    How, if possible, could I solve the equation: x''x=((x')^2)/2? Thanks.
  2. jcsd
  3. Mar 18, 2010 #2

    Filip Larsen

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    Gold Member

    If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.
  4. Mar 18, 2010 #3
    Try to find such a pair of real numbers a & b that d/dt (x^a x'^b) is proportional to x"x-((x')^2)/2.
  5. Mar 18, 2010 #4
    I've tried that but is there a way to actually solve it without guessing?
  6. Mar 18, 2010 #5


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    Equations like this, which do not contain the independent variable, are called autonomous systems. They are easily solved with the substitution x'' = x' dx'/dx . Then collect the terms in x' and x and integrate them both. This will give you x' in terms of x, which you can then integrate a second time.
  7. Mar 18, 2010 #6
    Do you only require a numerical solution given some particular boundry conditions?
  8. Mar 18, 2010 #7

    [tex] \frac{1}{2x}=\frac{\ddot{x}}{x}=-\frac{d}{dx}(\frac{1}{\dot{x}}) [/tex]

    Then, reduce to a quadrature:

    [tex]dt=(C-ln(\sqrt{x}))dx =d[x(C'-ln(\sqrt{x}))] [/tex]
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