x^y=y^x

x^x=y^(9y)

I need this in next few hours please help :(

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- Jan 23rd 2008, 04:11 PMAHDDMSolve System
x^y=y^x

x^x=y^(9y)

I need this in next few hours please help :( - Jan 23rd 2008, 04:38 PMmr fantastic
- Jan 23rd 2008, 05:02 PMAHDDM
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 - Jan 23rd 2008, 05:53 PMmr fantastic
- Jan 23rd 2008, 06:08 PMAHDDM
That looks all right to me but I just plugged in to check it doesnt come out

- Jan 23rd 2008, 06:44 PMmr fantastic
You're right. I should have remembered that the only integer solutions of are (2, 4) and (4, 2) .....

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

If you sub x = 3y into you get:

.

An*approximate*solution is . 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?