Euler theorem doubts

1. Oct 6, 2008

Nina2905

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

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. Oct 7, 2008

HallsofIvy

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