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Rayleigh's differential equation

  1. Dec 29, 2012 #1
  2. jcsd
  3. Dec 29, 2012 #2

    dextercioby

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  4. Dec 29, 2012 #3
    Well, i am asked to numerically solve it and produce a phase diagram.
    Should its value be given to me?
     
  5. Dec 29, 2012 #4

    dextercioby

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    I guess it should, so you're free to choose any value you want: Take [itex] \mu =1 [/itex] and solve it numerically.
     
  6. Dec 30, 2012 #5
    You're right , it was supposed to be given.
    Rayleigh's DE is [itex]y''-\mu y' + \frac{\mu (y')^3}{3} + y = 0[/itex]
    By rearranging it to a system of DEs, you get
    [tex]
    y_1 = y , y_1' = y_2 \\
    y_2' = \mu y_2 - \frac{\mu (y_2)^3}{3} - y_1
    [/tex]

    So i have only the derivative of y2 , i.e. the 2nd derivative of y1.
    Since i don't have an analytical description of y2 , how do i compute it with specific parameters, according to the numerical method.
    For example, for the classic Runge Kutta method,where f = y'
    [tex]
    k_1 = hf(x_n,y_n) = hy_2(n)\\
    k_2 = hf(x_n + 0.5h,y_n + 0.5k_1) = ?
    [/tex]
    I should numerically approximate the intermmediate values as well?
     
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