?Homogeneous Functions A function f is called homogeneous of degree n if it satisfies

Chapter 12, Problem 57

(choose chapter or problem)

Homogeneous Functions A function f is called homogeneous of degree n if it satisfies the equation

\(f(t x, t y)=t^n f(x, y)\)

for all t, where n is a positive integer and f has continuous second-order partial derivatives.

Show that if f is homogeneous of degree n, then

(a) \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\)

[Hint: Use the Chain Rule to differentiate f(t x, t y) with respect to t.]

(b) \(x^2 \frac{\partial^2 f}{\partial x^2}+2 x y \frac{\partial^2 f}{\partial x \partial y}+y^2 \frac{\partial^2 f}{\partial y^2}=n(n-1) f(x, y)\)