Does Euler's homogenous function theorem hold for functions homogenous in some of its independent variables? All proofs of Euler's homogenous functions theorem I've come across seem to assume the function is homogenous in all its independent variables. But does the theorem also hold in the case where the function is homogenous in a subset of its independent variables?

Yes. If a function is _weighted homogeneous_ , namely, if there are positive integers $q_1, \dots, q_n, d$ such that \begin{align} \lambda^{d} f(z_1,\dots,z_n) = f(\lambda^{q_1} z_1, \dots, \lambda^{q_n} z_n), \end{align} then it satisfies the _weighted Euler equation_ , \begin{align} d f = \sum_{i = 1}^{n} q_i z_i \frac{\partial f}{\partial z_i}. \end{align} For example, $f = x^{2} + y^{3} + z^{5}$ is weighted homogeneous with weights $q_1 = 15, q_2 = 10, q_3 = 6$ and weighted degree $d = 30$.