Parameterizing a function where you cannot isolate y If I'm given a curve as an implicit function, is there some algorithm for obtaining a parametrization of the curve?

For example, how would I parameterize

$\frac{2x}{y} - \frac{3y}{x} = 8$

I know I might use $\cosh$ and $\sinh$, but I'm wondering if there's a surefire way to parameterize a curve, when you have a formula for it like this.

I can't isolate y, clearly, but you can re-arrange it to

$ 2x^2 - 3y^2 - 8xy = 0 $

I'm hoping there's a way to find a trig rep for an arbitrary equation that I don't know about.

Where do you go from here?

Is it just trial and error / recognizing the form and choosing trig functions?
Solve a quadratic equation for $y$.
Quoting Arturo Magidin:


There is no general algorithm that will always yield a good parametrization of any curve. Relatively simple trigonometric parametrizations cannot work in general, because they always yield closed curves (being periodic). Parametrizing curves generally takes ingenuity, not unthinking application of general rules, though practice and familiarity with a variety of tricks will help (much as with integration)