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Math Help - Find and sketch region of existence and uniqueness

  1. #1
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    Find and sketch region of existence and uniqueness

    I have been asked to look at the following question over the weekend.

    The problem is I do not really understand how to start to tackle this problem.

    Consider the following equation y' = xLN(x+y)

    I've had calc 1,2,3 and linear algebra but for some reason I'm not sure how to start this problem. Not sure how to sketch it. It's been a while for me... I had linear in the spring of 2011, Calc 3 in the fall of 2010, Calc 2 in the spring of 2010 and Calc 1 in the fall of 2009.

    I appricate any pointers!
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  2. #2
    A Plied Mathematician
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    Re: Find and sketch region of existence and uniqueness

    Do you mean

    y'=x\ln(x+y)?

    If so, there's a well-known existence and uniqueness theorem that you should apply to this DE to answer your question.
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  3. #3
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    Re: Find and sketch region of existence and uniqueness

    That's exactly what I mean!

    I'll try to research for a existence formula for this question.
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  4. #4
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    Re: Find and sketch region of existence and uniqueness

    I have found the theorm though I'm not sure I fully understand it.

    I have taken the original formula and found the deriviate with respect to y to be

    df/dy = x/(y+x) by using the chain rule.

    y' = x(LN(X+Y)) => x * 1/(x+y) which simplifies to x / (y+x)

    sorry for not using latex. I will re learn how to use the tags so that it's done properly in the future.

    I'm not sure it's unique though? I know it's only zero when both X and Y are 0.

    This point is undefined as it would be 0/0 correct?

    so my original differential equations derivative is not continuous everywhere?
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  5. #5
    A Plied Mathematician
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    Re: Find and sketch region of existence and uniqueness

    The region x + y < 0 is not in the domain of your DE, since the logarithm function is not defined there. Rearranging, you get that y must be strictly less than -x. If it is true that y < -x, then your df/dy is differentiable in y, and continuous in x. Thus, by the Picard-Lindelöf Theorem, your solution exists and is unique. So, sketch that region, and you're done. Make sense?
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