Chapter 14.5, Problem 57E

Homogeneous Functions A function $f$ is called homogeneous of degree $n$ if it satisfies the equation $f(tx,ty)=t^{n}f(x,y)$ for all $t$ , where $n$ is a positive integer and $f$ has continuous second-order partial derivatives.

57. Show that if $f$ is homogeneous of degree $n$ , then

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

[Hint: Use the Chain Rule to differentiate $f(tx,ty)$ with respect to $t$ .]

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