Convert function $\ x^4 - y^4 = xy$ to a parametric form

Question

I can't figure out how to convert this function to parametric form.

$$\ x^4 - y^4 = xy$$

$$\ x(t) =? $$ $$\ y(t) =? $$

Any help would be greatly appreciated. Thanks!

Answer 1

Observe,

$$(x^2+y^2)(x^2-y^2) = xy\tag{1}$$

and let

$$x(t)=z(t)\cos t, \>\>\>\>\>y(t)=z(t)\sin t$$

Then, plug above form into (1) to obtain $z(t) = \frac{1}{\sqrt2}\sqrt{\tan 2t}$. Thus, the parametrized expressions are,

$$x(t)=\frac {1}{\sqrt2}\cos t\sqrt{\tan 2t} $$

$$y(t)=\frac {1}{\sqrt2}\sin t\sqrt{\tan 2t} $$

Answer 2

Substituting $$y=tx$$ we get $$x^4-t^4x^4=x^2t$$ so we get $$x^2(1-t^4)=t$$ for $$x\ne 0$$ and we have found $$x^2(1-t^4)=t$$