I am struggling to find a function $\varphi:\mathbb{R}^2\to \mathbb{R}$ such that:
$$ \frac{\partial \varphi}{\partial x}(x,y) = \frac{-x^2}{(1-x-y)^2} \\ \frac{\partial \varphi}{\partial y}(x,y) = \frac{-y^2}{(1-x-y)^2} $$
2026-05-04 15:44:20.1777909460
Solve analytically PDE rational functions
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No such function exists since ${\displaystyle {\partial \over \partial y} \frac{-x^2}{(1-x-y)^2} \neq {\partial \over \partial x} \frac{-y^2}{(1-x-y)^2}}$ but one always has ${\displaystyle {\partial \over \partial y} {\partial \phi \over \partial x} = {\partial \over \partial x} {\partial \phi \over \partial y}}$ by the equality of mixed partials.