Question

Solve `{(x^y=y^x),(x^2=y^3):}` ?

Answer 1

#((x=0, y=0), (x=1,y=1), (x=27/8, y = 9/4))#

Explanation:

`x^y=y^x` admits as solution `x=y`

Now applying `log` to the equations

`{(x^y=y^x),(x^2=y^3):}`-----(1)

we obtain

`{(y log x=x log y),(2 logx=3 logy):}`

Dividing term to term we obtain the relationship

`y/2=x/3`

The solutions for (1) are obtained solving

`{(x=y),(x^2=y^3):}`

and

`{(x^2=y^3),(y/2=x/3):}`

so they are

#((x=0, y=0), (x=1,y=1), (x=27/8, y = 9/4))#

Attached a figure showing the interceptions

In red the two leafs of `x^y=y^x`
blue dotted `x=y`
blue continuous `y/2=x/3` and
black `x^2=y^3`

The intersections have a black dot over.

The solution at `0,0` can be understood as a regularization.