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In my native language, we often talk about (what could be literally translated as) “modifying” or “refining” mathematical equations and expressions into a different form. Example sentences could be something like:

Refine this parametric line equation into the slope-intercept form.
Modify the integral into a simpler form.
Refine the polynomial equation to the factored form.

(All these “refinements” are implied to result in expressions which are equivalent to the original, yet simpler or having different qualities.)

I don't think I ever hear something like this in English, usually people use less direct expressions (“find the factored form“, without emphasis on the original expression) or more specific expressions (“factor the polynomial”, “substitute into the equation”). I often struggle to translate such sentences, because I often need to put emphasis on both the original expression being refined and the requested final form, but without being specific about the method.

Is any of the given sentences valid in English? Is there a natural way to say something like this?

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Your sentences are valid, but not the way that maths questions are usually asked.

It is probably a matter of "what we are used to", because I find "refine the parametric equation to cartesian form" to be far less direct than "Find the cartesian form...". Probably that is just because the English idiom is what I am used to.

So in English maths questions, we would rather say

  • Write this parametric line equation in slope-intercept form.
  • Simplify this integral
  • Factorise this polynomial equation.

For me, these are simple and direct. They all use the imperative form of the verb.

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