How to Solve This: yˣ= xʸ Definitively

This is not a new mathematics problem, but it always arouses curiosity. So let’s see how we can solve it by a very robust methodology.

The trivial solution, of course, would be to make x=y. But let us prove it by analytical arguments.

We can rewrite the expression in the form:

and consider as an implicit function

where to apply the system of equations

which results in

such a system of equations leads us to the following expression

whose solution results in

This is a necessary but not sufficient solution. Therefore, other solutions need to be found. We can rewrite the equation in the form:

and from there, from the first equation obtain another one explicitly

thus,

which we can rewrite in the form

or

considering x=y as a boundary condition, then we can write

and using that, we do

So we can perform the following algebraic steps

and so, we get c:

Therefore, the other solutions to the question will be

and

The solution we have just solved is the set of points that form a locus of an implicit function and its graph in the Cartesian plane resembles a “trident”.