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General solution of ODE

  1. Jun 23, 2009 #1
    The flow defined by a differential equation/general solution

    I'm trying to find the 'general solution'(perhaps more accurately the flow)
    of the following ODE:

    [tex]x' = x(1-x)[/tex]

    i.e. [tex]\lambda_{max}(t, \tau,\xi)[/tex] and the domain where it's defined [tex]\Omega[/tex]

    note:
    The definition of 'general solution' i'm refering to here is the following:

    Given a open [tex]D\subset R^{1+N} [/tex], a continous and with respect to x Lipschitz-continous function [tex]f:D \rightarrow R^{N} [/tex], and the differential equation

    [tex]x' = f(t,x)[/tex]

    The function [tex]\lambda(t, \tau,\xi) := \lambda_{max}(t, \tau,\xi)[/tex] defined for [tex](t, \tau,\xi)\in\Omega := \left\{(t, \tau,\xi)\inR^{1+1+N} : (\tau,\xi)\in D, t \in I_{max}(\tau,\xi)\right\}[/tex]
    is called the 'general solution' of the differential equation.

    [tex]\lambda_{max}[/tex] is the maximal solution, and [tex]I_{max}(\tau,\xi)[/tex] the maximal interval of
    existence of the solution of the intial value problem [tex]x(\tau)=\xi[/tex]

    Any recommended examples/literature?
     
    Last edited: Jun 23, 2009
  2. jcsd
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