Solving Euler Theorem Doubts with Partial Derivatives

[Nina2905]

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 got to 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...

 

[HallsofIvy]

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.