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Homework Help: Solving non-linear second order ODE

  1. May 18, 2010 #1
    1. The problem statement, all variables and given/known data

    It will be great if someone could show me some options:
    I need to prove the following:

    A particle of mass m is attracted toward a fixed point O (the origin) with a force proportional
    to its instantaneous distance from O raised to a positive integer power, i.e., according to the force
    law F (x) = −k(x^n) . (Note that n must be odd in order for F (x) to be an attractive force!) Suppose
    that, at time t = 0, the particle is released from rest at x = x0 . Show that the time T for the particle to reach O is given by the formula:
    T = C / ( (xo)^(n-1)/2 )

    2. Relevant equations

    Determine explicitly the coefficient C in terms of (i) the Beta function and (ii) Gamma functions

    3. The attempt at a solution

    i tried solving it manually using definite integrals and arrived at
    C = (m(n+1)/(2*K))^-1/2

    but this looks nothing like the gamma or beta function
    also, I was unable to find a normal method to solve the 2nd order non linear ODE:
    x'' = (-k/m)x^n

    because of the x^n... all common methods are give for just x or with some x'
    Also tried series expansion solution to the ODE... but the t was in several pieces and cannot put it together in the required form

    Also tried using Emden-Fowler equation but arrived to a very similar value of C without any relation to the beta or gamma

    Any help will be greatly appreciated
     
  2. jcsd
  3. May 18, 2010 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    This is fairly straightforward by use of the chain rule:

    [tex]\ddot{x}=\frac{d}{dt}\dot{x}=\frac{dx}{dt}\frac{d}{dx}\dot{x}=\frac{1}{2}\frac{d}{dx}\dot{x}^2[/tex]

    Which gives you a separable ODE:

    [tex]\frac{1}{2}\frac{d}{dx}\dot{x}^2=-\frac{k}{m}x^n[/tex]
     
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