Question

form the partial equation by eliminating the function f from z=f(y/x)

Answer 1

The equations can be obtained in different ways. For example, we can consider derivatives: $z_y=f'\frac{1}{x}$ and $z_x=-\frac{y}{x^2}f'$. Then, the following equation can be made: $z_y+\frac{x}{y}z_x=0.$ It is equivalent to: $yz_y+xz_x=0$, except the point $y=0.$