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Help with pertubation method

  1. May 22, 2009 #1


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    Suppose I have a equation of the form

    F(x, k) = G(x, k)

    which is unsolvable analytically. We apply the method of pertubtaion (k is small quantity) and let the first order solution approximated as

    [tex]x = x^{(0)} + kx^{(1)}[/tex]

    we need to put this back to the equation and find out [tex]x^{(1)}[/tex]. If after substitution, F(x, k) will only give constant term or term with k and G(x, k) will give some extra terms of order [tex]k^2[/tex]. For example,

    [tex]F(x, k) = ax^{(0)} + bkx^{(1)}[/tex]
    [tex]G(x, k) = cx^{(0)} + dkx^{(1)} + h k^2 (x^{(0)}-x^{(1)})[/tex]

    shall I directly drop the high-order term or let the coefficient of the high-order term to zero? For the later one, we will introduce another condition to solve the equation (seems not correct!?)
  2. jcsd
  3. May 23, 2009 #2


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    Science Advisor

    When you set x= x(0)+ kx(1), you are saying that the perturbation (measured by k) is small enough that higher powers can be ignored. You drop any term with k2 or higher.
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