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Thread: Uniqueness Theorem

  1. #1
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    Uniqueness Theorem

    Hi all,

    I have the following D.E.

    $\displaystyle \frac{dy}{dt} = \sqrt{y^2+1},\\ y(t_0) = y_0$

    How can i find a value of $\displaystyle y_0$ and a value of $\displaystyle t_0$ such that there is a unique solution to the initial-value problem?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    The functions

    $\displaystyle f(t,y)=\sqrt{y^2+1},\;\;\dfrac{\partial f}{\partial y}$

    are continuous on $\displaystyle \mathbb{R}^2$ , so for every $\displaystyle (t_0,y_0)\in \mathbb{R}^2$ there exists a unique solution satisfying $\displaystyle y(t_0)=y_0$ .
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