Solving non linear system of equation

Question

Im wondering how to solve this system of equations?

$$\frac{2xy}{(1-x^2-y^2)^2}=0$$ $$\frac{-x^4-2x^2y^2+3x^2-y^4+y^2}{(1-x^2-y^2)^2}=0$$

Since they both have the same denominator maybe it's a good ide to use elimination? Or is there an easier approach to this problem?

Answer 1

Hint: The denominator is not allowed to be zero. For what values $(x,y)$ will the denominator become zero? All zeros lie on a circle around the origin with radius $1$. Can you formalize this? Let us call this set $D$ and the complementary set as $\bar{D}$.

Edit: In order to find the zeros of the denominator we set the denominator to zero.

$$(1-x^2-y^2)^2=0 \implies 1-x^2-y^2=0 \implies 1^2=x^2+y^2$$

The last expression describes a circle with radius $r=1$ which is centered at $(x_0,y_0)=(0,0)$. See the wiki article for more information.

Then assume that $(x,y) \in \bar{D}$. Then you can multiply with the denominator, because it is never zero, after excluding all possible zeros.

Can you solve the system after this simplification? Use the first equation $xy=0$ as a starting point.

Answer 2

Multiplying both equations with the denominators will give you two equations in polynomials with variables $x$ and $y$.

Solve these as you normally would. You then want to check for which of these solutions the denominator does not vanish.