I finally solved the problem

, and here is the solution:

First, if we need to find the first partial derivatives of x and y with respect to u and v, given 2u - v + x^2 + x*y = 0, u + 2v + x*y - y^2 =0 it is important to be clear that in this case u and v are

__independent __variables and x and y are the

__ dependent __ones. This seems obvious, of course, but when I was actually trying to solve the problem, I wasn't correctly considering it. So:

1. Differentiate the first equation with respect to u: 2 + 2x*∂x/∂u +y*∂x/∂u +x*∂y/∂u = 0 2. Differentiate the second equation with respect to u: 1 + y*∂x/∂u +x*∂x/∂u- 2y*∂y/∂u = 0; from here get ∂y/∂u = (1 + y*∂x/∂u)/(2y-x), replace ∂y/∂u in the first equation: 2 + 2x*∂x/∂u +y*∂x/∂u +x*(1 + y*∂x/∂u)/(2y-x) = 0; From here it is easy to see that, indeed, ∂x/∂u = (4y-x)/2(x^2 - 2xy - y^2). It turned out to be not such a hard problem (it always seems to be easy once it is solved

)