Answer to Question #105495 in Differential Equations for khushi

Question

State whether the following statements are true or false. Give reasons for your answers (2) The function f(x, y) =max{y/x, x} is a homogenous function on R^2

Answer 1

The definition of a homogeneous function states ∃n∈Z , ∀v∈R^2, ∀a∈R (f(av)=(a^nf)(v))⇔(f is homogeneous).

For our function, this translates to ∃n∈Z , ∀f(x,y)∈R^2, ∀a∈R (f(ax, ay))=(a^n)f(x,y))⇔(f is homogeneous).

In this case, f(ax, ay)= max(ax/ay, ax) which simplifies to max(x/y, ax).

Different inequalities for x, y, and a will result in either n=1 or n=0, since one value of n does not apply to all scenarios, f is not a homogeneous function.