1. ## partial differential equation

A function $\displaystyle f$ of two variables is said to be homogeneous of degree n if $\displaystyle f(tx,ty)=t^nf(x,y)$ whenever $\displaystyle t>0$.

Show that such a function $\displaystyle f$ satisfies the equation $\displaystyle x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf$

2. Originally Posted by vuze88
A function $\displaystyle f$ of two variables is said to be homogeneous of degree n if $\displaystyle f(tx,ty)=t^nf(x,y)$ whenever $\displaystyle t>0$.

Show that such a function $\displaystyle f$ satisfies the equation $\displaystyle x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf$

Derive both sides of $\displaystyle f(tx,ty)=t^nf(x,y)$ wrt $\displaystyle t$ and then evaluate the result at $\displaystyle t=1$ .

Tonio

3. Originally Posted by tonio
Derive both sides of $\displaystyle f(tx,ty)=t^nf(x,y)$ wrt $\displaystyle t$ and then evaluate the result at $\displaystyle t=1$ .

Tonio
Sorry i still dont fully understand. Can you explain this in more detail. Also im not sure whether the f on the RHS means f(x,y) or f(tx,ty)

4. Originally Posted by vuze88
Sorry i still dont fully understand. Can you explain this in more detail. Also im not sure whether the f on the RHS means f(x,y) or f(tx,ty)

It is exactly as I wrote it: derivate the left side wrt t, derivate the right side wrt t, they're equal, of course, and now put t=1 .

Tonio

5. lol yeah but how to i differentiate with respect to t when the function is in terms of tx and ty

6. Use the chain rule, of course!

$\displaystyle \frac{df(ax)}{dx}= a\frac{df}{dx}$

7. ## Euler's theorem

Originally Posted by HallsofIvy
Use the chain rule, of course!

$\displaystyle \frac{df(ax)}{dx}= a\frac{df}{dx}$
this is what Euler's theorem on partial derivatives says.