Technique for simplifying partial derivatives of homogeneous functions

[Mr.Rockwater]

Homework Statement

I need to prove that $x\frac{ \partial^2z}{ \partial x^2} + y\frac{\partial^2z}{\partial y\partial x} = 2\frac{\partial z}{\partial x}$

Homework Equations

$z = \frac{x^2y^2}{x+y}$

The Attempt at a Solution

I actually did it the long way and I got the right answer but here is my teacher's solution :

$z = \frac{x^2y^2}{x+y}$

$\Rightarrow x\frac{ \partial z}{ \partial x} + y\frac{ \partial z}{ \partial x} = 3\frac{ \partial z}{ \partial x}$

$\Rightarrow \frac{ \partial z}{ \partial x} +x\frac{ \partial^2z}{ \partial x^2} + y\frac{ \partial z}{ \partial x} = 3\frac{ \partial z}{ \partial x}$

Answer follows.

To be honest, I have absolutely no idea about what technique he actually uses there. Is there any "rule" or "trick" that I am not aware of here?

 

[lurflurf]

It follows from observing z is homogeneous of degree 3, Euler's homogeneous function theorem, and interchanging operators.

x z_xx+y z_yx=(x z_x+y z_y-z)_x
by commuting operators
=(3-1)z_x=2z_x
by Euler's homogeneous function theorem
thus
z_x is homogeneous of degree 2
or we could go backwards and just show z_x is homogeneous of degree 2

 

[Mr.Rockwater]

Thank you! Our teacher didn't ever mention homogenous functions though, I assume this ain't going to be in the exam. At least I'll have that tool in my arsenal