Solve this

[faint grove]

2x+x^2y = y
2y+y^2z = z
2z+z^2x = x
find x y z

[thin wadi]

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[hybrid flare]

Probably worth adding up all three equations first.

[bleak thistle]

are you quite sure this is correct
the solutions to this are very complicated

and there are like 9

[bleak thistle]

yep it seems right

are you allowed a calculator or anything

[faint grove]

its fine

[bleak thistle]

notice that if you solve for y in terms of x, you can write y = f(x)
and z = f(y)
and x = f(z)
then you finally get x = f(f(f(x)))

at this point it becomes a very horrible rational function

[hybrid flare]

I still think adding them first might produce something nice.

Well, maybe...

[flat compass]

note that

$$ y= \frac{2x}{1-x^2} = \frac{2\tan s}{1-\tan ^2s} = \tan 2s $$

[hybrid flare]

Ohh.

Nice idea!

[flat compass]

playing around with this, we get (x,y,z) in terms of tangents

[flat compass]

this is also a nice idea, putting f(s) = 2s / (1-s^2) so we have x = f^3(x)

[bleak thistle]

ehehh

yours is much better

[hybrid flare]

So, if I understand correctly, we eventually arrive to ||tan(s) = tan(8s)||?

[flat compass]

that's right so 7s = n*pi is implied

[versed basin]

should be 2tan(s) on the numerator

[whole flint]

aL

[flat compass]

you're right, thanks I did the computation with the correct expression

[hybrid flare]

I see. That's really cool! Would've taken me a while to attempt this.

[frigid surge]

@flat compass has given 1 rep to @versed basin

[versed basin]

urwelcome

[faint grove]

thanks for ideas guys 🙂

[bleak thistle]

can the tan solution find the (non real) solutions where x = y = z

[flat compass]

over R, yes, haven't thought about C

[bleak thistle]

it seems like some infinity gets involved

[bleak thistle]

is real values implied

[faint grove]

+close