Algorithm which adapt (solve) the complex equations ( implicit function f(x,y) )

Question

I\'m trying to adapt some equations (implicit f(x,y)) in order to be able list the Y for corresponding X-Value.
The equations could be e.g. as follows:

y^2 =         


        

Answer 1

I'd like to point out that this problem cannot be solved exactly in general. The cited solution for the quadratic case (y^2) can be extended to the cubic case and quartic case (there are a general complicated solutions). But there is a math theorem (from Galois theory) that states that there is no general solution for the quintic equation (and so on). In your case, maximum degree is 3, so you can use the cubic equation from wikipedia. For the heart graphic write: x^2*y^3 - y^2 -(x^2-1) = 0 and treat x as constant. For the sqrt case, get rid of it. Square both sides of equation, isolate y and you end up with a quartic equation on y, that you can solve using wikipedia's quartic equation knowledge.

Anyway, if you don't have a very strong reason to do this, don't do it, as the computer can solve this numerically for you. Standard approach is to calculate this implicitly, as in the plots you made.

I hope this helps.