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Trouver les applications \(f : {(\R^{+*})^2} \to \R\) de classe \(\mathcal{C}^2\) vérifiant : \(x^2\frac{\partial^2 f}{\partial x^2} + 2xy\frac{\partial^2 f}{\partial x \partial y} + y^2\frac{\partial^2 f}{\partial y^2} = 0\). On utilisera le changement de variables : \(u = xy\), \(v = \frac xy\).