Solution of a system of functional-differential equations arising from an optimization problem (Q1074851)

The following problem arises in connection with an optimization problem: Let I be an interval in [0,\(\infty)\). Characterize the functions \(f: I^ n\to {\mathbb{R}}\) with continuous, nonzero partial derivatives on \(I^ n\) which satisfy \(f_{x_ i}/f_{x_ j}=g_{ij}(x_ i/x_ j),\) \(1\leq i\), \(j\leq n\), \(n>2\). That is, which functions f have quotients of partials equal to some function of the quotient of the respective variables? It is proved that the functions f have the form \(f(x_ 1,...,x_ n)=H(\alpha_ 1 x_ 1^{\beta}+... +\alpha_ n x_ n^{\beta})\quad,\) and \(f(x_ 1,...,x_ n)=H(a_ 1 \ln x_ 1+...+a_ n \ln x_ n)\) where H is any continuously differentiable function with nonzero derivative and \(\alpha_ 1,...,\alpha_ n\), \(\beta\), \(a_ 1,...,a_ n\) are any nonzero constants.