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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?

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tomatopipps
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1 Answers1

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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 $t>0$, $t\neq 1$.

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