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Topic: Powerful Equation (Read 530 times) |
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ThudnBlunder
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Powerful Equation
« on: Jan 20th, 2009, 5:16pm » |
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If x  y, find all real solutions to xy = yx
Which solutions are rational? Which are integral?
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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balakrishnan
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Re: Powerful Equation
« Reply #1 on: Jan 21st, 2009, 3:24am » |
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The equation can be written as log(x)/x=log(y)/y.
It is an increasing function from 0 to e and decreases to zero from e to  . So if x and y has to be distinct, then x and y must lie in the range (1,). One of the root will lie in the range (1,e) and the other in the range (e,) .
If x and y are integers, then one of them must be 2(since there is only 1 integer in the range (1,e)) and the other is 4.
So the integer solutions are x=2,y=4 and x=4,y=2.
For rational solutions, it is not very difficult to see that x and y would take the form
x=((n+1)/n)^n, y=((n+1)/n)^(n+1) or
x=((n+1)/n)^(n+1), y=((n+1)/n)^(n) or
where n is any integer
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| « Last Edit: Jan 21st, 2009, 3:24am by balakrishnan » |
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