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Math Help - Non-vanishing Solution in Differencial Equation

  1. #1
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    Non-vanishing Solution in Differencial Equation

    Say we wish to solve y' = y on (-\infty,\infty)*.

    1) y is non-vanishing i.e. y(\xi) \not = 0 for any \xi \in \mathbb{R}.
    Then we can write y'/y = 1 and thus (\log |y|)' = 1.
    This leads to \log |y| = x + k_1 \implies y = ke^x with k\not = 0

    2) y is the zero function but that is definitely a solution.

    3) y vanishes at some point i.e. y(\xi) = 0 for some \xi \in \mathbb{R}.
    This is were the difficultly is. If it vanishes at \xi and in the open interval I containing \xi then I think this leads to y being the zero function**. However, if y does not vanish on any open interval containing it then it means there is a point \xi_1 such that y(\xi_1) \not = 0. But by continuity it means there is an interval J containing \xi_1 such that y is non-vanishing on J. By applying #1 to this restricted function, y|J, we find that y|J must be of the form ke^x,k\not = 0. And this perhaps shows that y is ke^x everywhere.*** Therefore y is non-vanishing everywhere. A contradiction if y is a non-zero function. Thus, this case does not exist for non-zero functions.

    How do we complete the proof?
    And is there a general theorem that tells us we can divide the functions we are solving for without worrying it can vanish.


    *)This can be solved without seperation of variables i.e. with integrating fractor.
    But with this something interesting happens.

    **)Though I did not find the complete proof. I guess the proof comes down to a analogue of the identity theorem from complex analysis applied to real functions. This is because the solution to the differencial equation is \mathcal{C}^{\infty} and we can possibly even proof its analyticity. With analyticity we can have an analogue of the identity theorem.

    ***)Again by identity theorem for real function - by using power series.
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  2. #2
    Super Member Rebesques's Avatar
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    This is a homogeneous linear equation, so by the uniqueness theorem the only solution that can vanish at any point is y=0.
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