Déterminer les fonctions \(f : {D \subset {\R^2}} \to \R\) vérifiant :
1
\(\begin{cases} \frac{\partial f}{\partial x} = \frac {2+x}y \cr \frac{\partial f}{\partial y} = \frac {2+y}x \cr\end{cases}\)
2
\(\begin{cases} \frac{\partial f}{\partial x} = \frac {1-y}{(x+y+1)^2} \cr \frac{\partial f}{\partial y} = \frac {2+x}{(x+y+1)^2} \cr\end{cases}\)
3
\(\begin{cases} \frac{\partial f}{\partial x} = \frac{y^2}{(x+y)^2} \cr \frac{\partial f}{\partial y} = \frac{x^2}{(x+y)^2} \cr \end{cases}\)
4
\(\begin{cases} \frac{\partial f}{\partial x} = 2x + \frac 1y \cr \frac{\partial f}{\partial y} = 2y - \frac x{y^2} \cr \end{cases}\)