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Sum of partial derivatives

  1. Oct 29, 2011 #1
    1. The problem statement, all variables and given/known data
    I need to prove that [itex]x\frac{ \partial^2z}{ \partial x^2} + y\frac{\partial^2z}{\partial y\partial x} = 2\frac{\partial z}{\partial x}[/itex]

    2. Relevant equations

    [itex] z = \frac{x^2y^2}{x+y} [/itex]

    3. 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 :

    [itex] z = \frac{x^2y^2}{x+y} [/itex]

    [itex]\Rightarrow x\frac{ \partial z}{ \partial x} + y\frac{ \partial z}{ \partial x} = 3\frac{ \partial z}{ \partial x} [/itex]

    [itex]\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} [/itex]

    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?
     
  2. jcsd
  3. Oct 30, 2011 #2

    lurflurf

    User Avatar
    Homework Helper

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

    x zxx+y zyx=(x zx+y zy-z)x
    by commuting operators
    =(3-1)zx=2zx
    by Euler's homogeneous function theorem
    thus
    zx is homogeneous of degree 2
    or we could go backwards and just show zx is homogeneous of degree 2
     
  4. Oct 31, 2011 #3
    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 :tongue:
     
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