Let $X=(x_0,y_0)=(1,1) \in \mathcal{R}^2$. Prove that there exists $\rho>0$ and continuously differentiable functions $u,v,w: B_{\rho}(X) \to \mathcal{R} $ such that
$u(X) = 1, v(X) = 1, w(X) = -1$
and
$u^5(x,y) + xv^2(x,y)-y+w(x,y) = 0$
$v^5(x,y)+yu^2(x,y)-x+w(x,y)=0$
$w^4(x,y)+y^5-x^4=1$
Can you give me any hint to solving this or how to approach this? I think it has something to do with implicit functions but I am not sure. Thank you.