This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos relationship.
I want to find a parametric version of this: $\space x^y=y^x$
I've already found the answer on this site, but it doesn't show a method for getting it.
Would anyone mind giving a little tutorial?
One standard way to find a parametrization of $F(x,y)=0$ is to set $y=tx$. In your case it gives the following parametrization of non-trivial branch: $$\Large x=t^{\frac{1}{t-1}},\quad y=t^{\frac{t}{t-1}},$$ for $t>0$, $t\neq 1$.