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Euler theorem doubts

  1. Oct 6, 2008 #1
    firstly, all d's i use will mean partial derivative 'do' because i don't have the font installed. sorry :(

    please help me with these.. u gotta use euler theorem
    1. if z= xf(y/x) + g(y/x), show that x2(d2z/dx2) + 2xy(d2z/dxdy) + y2(d2z/dy2) =0
    2. if z= (xy)/(x-y), PT (d2z/dx2) + 2(d2z/dxdy) + (d2z/dy2) = 2/(x-y)

    thanks...
     
  2. jcsd
  3. Oct 7, 2008 #2

    HallsofIvy

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    First, it's not a matrer of having "fonts" installed, just use LaTex with [ tex ] and [ /tex ] (without the spaces) beginning and ending. To see LaTex commands, click on any formula on this board.

    I'm not sure which "Euler Theorem" you mean (there are many). It looks to me like like you only need to differentiate.

    If z= xf(y/x)+ g(y/x), then
    [tex]\frac{\partial z}{\partial x}= f(y/x)+ x f'(y/x)(-y/x^2)+ g'(y/x)(-y/x^2)[/tex]
    by the chain rule. Doing that again,
    [tex]\frac{\partial^2 z}{\partial x^2}= [f'(y/x)(-y/x^2)]+ [f'(y/x)(-y/x^2)+ xf"(y/x)(-y/x^2)^2+ xf'(y/x)(2y/x^3)]+ g"(y/x)(-y/x^2)^2+ g'(y/x)(-2y/x^3)][/tex]
    Of course, that can be simplified a lot.
     
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