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Need help with ODE and Existence and Uniqueness Thm

  1. Jun 3, 2006 #1


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    Need help with ODE and "Existence and Uniqueness Thm"

    I'm currently helping my neice study for her exams and going though last years test there was this one question that I wasn't sure about.

    Considering the initial value problem,

    [tex]\frac{dy}{dx} = f(x,y)[/tex]

    Where [tex]f(x,y) = (1+x) \sqrt{y}[/tex] and y(1)=0.

    Does the "Existence and Uniqueness Theorem" guarantee the existence of a unique solution?

    I'm pretty sure that the theorem they are referring to is this one,
    See also,
    http://www.utpb.edu/scimath/wkfield/mod3/Exuni.htm [Broken]

    I'm thinking that it (the Theorem) doesn't guarantee a unique solution because [tex]\frac{\partial f}{\partial y}[/tex] is not well defined at the initial condition y=0.

    Unfortunately I'm not too familiar with this "Existence and Uniqueness" theorem. Also the DE does seem to have a well defined solution of y(x)=0 for x>=1, so I'm really not sure. Can anyone help me out here, thanks.
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jun 3, 2006 #2


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    you are quite right; the theorem cannot be used here to guarantee a unique solution.
    In this case, you have (at least) two solutions fulfilling the initial value problem:
    [itex]y_{1}(x)=0, y_{2}(x)=\frac{1}{16}((1+x)^{2}-4)^{2}[/itex]
    Last edited: Jun 3, 2006
  4. Jun 3, 2006 #3


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    Thanks for the answer arildno. I can see how you got the two solutions, y=0 by inspection and the other y = 1/16 (x^2 + 2x -3)^2 by seperating and integrating. Yes, now the question makes sense and it nicely shows the theorem in action. Thanks again :)
    Last edited: Jun 3, 2006
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