- #1

kostoglotov

- 234

- 6

## Homework Statement

Show that if f is homogeneous of degree n, then

[tex] x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) [/tex]

Hint: use the Chain Rule to diff. f(tx,ty) wrt t.

**2. The attempt at a solution**

I know that if f is homogeneous of degree n then [tex] t^nf(x,y) = f(tx,ty) [/tex]

But I'm really at a conceptual loss here.

I've tried the hint, I let f(tx, ty) = f(a,b) so a = tx and b = ty then

[tex] \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}\frac{\partial a}{\partial t}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial t} \\

\frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}x + \frac{\partial f}{\partial b}y [/tex]

And I have tried looking at

[tex] \frac{d}{dt}t^nf(x,y) = nt^{n-1}f(x,y) [/tex]

But beyond this I just cannot see what I need to do.

Thanks.