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Existence and Uniqueness of a solution for ordinary DE

  1. Sep 11, 2007 #1
    I just dont understand the idea behind it. I hate it when they throw these theories at us without proofs or elaborate explanations and just ask us to accept and applym mthem. Anyone care to enlighten me?
  2. jcsd
  3. Sep 12, 2007 #2
    Well, sometimes it's hard (or annoying) to find solutions to ODEs by hand and, hence, it is easier to show that there exists a (unique) solution to an initial value problem depending on the information you're given.
    However, in a lot of cases, existence/uniqueness is established in an attempt to formulate a general setting in which you are guaranteed to have a solution to an ODE, which is unique.
    For example, the classical analysis problem: Show that there exists a solution to [tex]y'=f(x,y)[/tex], [tex]y(0)=y_0[/tex], [tex]\|f(x,y)-f(x,z)\| \le k\|y-z\|[/tex] in some interval [tex][0,a][/tex], where [tex]x\in[0,a][/tex], [tex]y:[0,a]\rightarrow\mathbb{R}[/tex] and [tex]f \in C([0,a]\times\mathbb{R})[/tex].
    This could be formulated as a fixed point problem-- i.e. fixed points of:
    [tex]Ty=y_0 + \int_{0}^{x}f(t,y(t))dt[/tex], [tex]x\in[0,a][/tex], would satisfy the ODE above.
    Hence, one may use the Banach contraction mapping principle after showing that [tex]Ty[/tex] is a contraction mapping (in particular instances), and hence conclude that there exists a unique solution [tex]y[/tex] to the problem [tex]Ty=y[/tex] and hence, existence and uniqueness is established for this class of ODEs. (You would end up with some sort of restriction on [tex]a[/tex] though)
    I have no idea why you would be exposed to these sorts of ideas without proofs, since that would sort of defy the purpose.
    Last edited: Sep 12, 2007
  4. Sep 12, 2007 #3
    Quite often, in a physics course, uniqueness of the solutions to Poisson's equation or Laplace's equation are used to find solutions based on guesses and to justify the method of images. Proof is usually deferred.
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