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Differential Equation

  1. Feb 6, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex]
    \frac{dx}{dt} =\frac{ -x}{(t-1+e^{-x})}
    [/tex]

    Show that an approximate solution leads to,

    [tex]
    \frac{dx}{dt} = -\frac{ 1}{1-c1} [c1+(c2 + \frac{c2-c1/2}{1-c1})*t + O(t^3)]
    [/tex]




    2. Relevant equations



    3. The attempt at a solution

    The first equation is not separable.

    To approximate, assume
    [tex]
    x = c1*t+c2*t^2 + O(t^3)
    [/tex]
    Hence
    [tex]
    dx/dt = c1 + 2*c2*t + O(t^2).
    [/tex]

    If I equate

    [tex]
    dx/dt = c1 + 2*c2*t + O(t^2).
    [/tex]

    and

    [tex]
    \frac{dx}{dt} =\frac{ -x}{(t-1+e^{-x})}
    [/tex]

    Should I immediately taylor expand the exponential?
     
  2. jcsd
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