Approximations to the Roots of a Function

Question

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it when you have a function in two variables. The function I'm looking at is y^3+y +xy-x^3-2=0. I would appreciate any help.

Answer 1

There aren't any specific distinct "roots". The set of points $(x,y)$ where $y^3+y +xy-x^3-2=0$ is an entire curve in the $xy$ plane. Here's what the curve looks like:

You have to decide which point of that curve you want, by providing a second equation, for example. Then you can use Newton's method to solve for $x$ and $y$.

If you don't care which point of the curve you get, then $(1,1)$ is a nice one.