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Homework Help: Nonlinear differential equation

  1. May 9, 2012 #1
    1. The problem statement, all variables and given/known data

    In one problem I had got to this equations, but I was not able to solve it, because I'm actually
    on high school.

    The equation :
    d^2/dt^2(x) = -h*g/(h+x)

    I tried use separation of variables but I was not able to use the chain rule.
    Can anybody show me the steps with explanation?
    Last edited by a moderator: May 9, 2012
  2. jcsd
  3. May 9, 2012 #2


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    welcome to pf!

    hi mbadin! welcome to pf! :smile:

    (try using the X2 button just above the Reply box :wink:)
    either multiply both sides by dx/dt

    or use d2x/dt2 = vdv/dx

    (where v = dx/dt … you can prove it using the chain rule :wink:)
  4. May 9, 2012 #3


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    Gold Member

    OK, here is how your equation appears to me:
    [tex]\frac{d^2x}{dt^2} = -\frac{hg}{h+x}[/tex]
    If i understood your question correctly, then you need to integrate twice to get rid of the differentiation.
  5. May 9, 2012 #4


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    Science Advisor

    It's not that easy because the right side, that you want to integrate with respect to t depends upon the unknown function x.

    I would do what tiny-tim suggested: Let v= dx/dt so that [itex]d^2x/dt^2= dv/dt[/itex] and then, by the chain rule,
    [tex]\frac{dv}{dt}= \frac{dv}{dx}\frac{dx}{dt}= v\frac{dv}{dx}[/tex]

    Your equation becomes
    [tex]v\frac{dv}{dx}= -\frac{hg}{h+ x}[/tex]
    which can be integrated as
    [tex]\int v dv= hg\int \frac{dx}{h+ x}[/tex]

    It might well give you a function v= dx/dt that is difficult to integrate but that is the most direct method to solve this equation.
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