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Math Help - Solve System

  1. #1
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    Solve System

    x^y=y^x
    x^x=y^(9y)

    I need this in next few hours please help
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  2. #2
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    Quote Originally Posted by AHDDM View Post
    x^y=y^x
    x^x=y^(9y)

    I need this in next few hours please help
    (x^y)^x = (y^x)^x \Rightarrow x^{yx} = y^{x^2}.

    (x^x)^y = (y^{9y})^y \Rightarrow x^{xy} = y^{9y^2}.

    Therefore y^{x^2} = y^{9y^2} \Rightarrow x^2 = 9y^2.

    Therefore ........
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  3. #3
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    x^2=9y^2

    y=x/3

    x^x=y^(9*x/3)
    x^x=y^(3x)

    I got this so far<

    When I use x=3y and plugin to first equation

    I get to 3y^y=y^(3y)

    I did it and i got x=0 and y=0
    Last edited by AHDDM; January 23rd 2008 at 04:49 PM.
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  4. #4
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    Quote Originally Posted by AHDDM View Post
    x^2=9y^2

    y=x/3

    x^x=y^(9*x/3)
    x^x=y^(3x)

    I got this so far<

    When I use x=3y and plugin to first equation

    I get to 3y^y=y^(3y)
    x^2 = 9y^2 \therefore x = \pm 3y.

    Case 1: x = 3y.

    (3y)^{3y} = y^{9y} \Rightarrow 3^{3y} y^{3y} = (y^{3y})^2 \Rightarrow y^{3y} \left( 3^{3y} - y^{3y} \right) = 0 \Rightarrow 3^{3y} - y^{3y} = 0 \Rightarrow 3^{3y} = y^{3y}. So one obvious solution is y = 3 \Rightarrow x = 9 .......
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  5. #5
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    That looks all right to me but I just plugged in to check it doesnt come out
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  6. #6
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    Quote Originally Posted by AHDDM View Post
    That looks all right to me but I just plugged in to check it doesnt come out
    You're right. I should have remembered that the only integer solutions of x^y = y^x are (2, 4) and (4, 2) .....

    So (9, 3) is a spurious solution ......

    If you sub x = 3y into x^y = y^x you get:

    (3y)^y = y^{3y} \Rightarrow 3^y y^y = (y^y)^3 \Rightarrow y^y \left( 3^y - (y^y)^2 \right) = 0 \Rightarrow 3^y - y^{2y} = 0.

    An approximate solution is y = 1.73205 \Rightarrow x = 5.19615. This solution does seem to work ..... To express this solution for y in exact form, I think you'd need to use the Lambert W-function ......

    I haven't looked at the x = -3y case yet, but there will probably be similar if not worse difficulty in store ......

    Are you expected to get exact solutions?
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