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Math Help - partial differentiation

  1. #1
    Junior Member
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    Aug 2009
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    partial differentiation

    hello,

    i am asked to prove that if z= [f(y/x)]/x then

    ...pls read [z_x] as the partial of z w.r.t.x...

    x[z_x] + y[z_y] + z = 0 (*)

    i have tried totally differentiating

    dz = {-[1/x^2][f(y/x)] + [1/x][f_x] }dx + [1/x][f_y]dy

    but then i get from the LHS of (*) that [1/x][f_x] }dx + [1/x][f_y]dy = ?

    I am given clues for two methods that i may follow:

    1) set y/x =v

    and

    2) to use the fact that for a homogenous function g(x,y) of nth order it is true that x[g_x] + y[g_y] = 0


    i am stuck
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  2. #2
    Banned
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    Quote Originally Posted by pepsi View Post
    hello,

    i am asked to prove that if z= [f(y/x)]/x then

    ...pls read [z_x] as the partial of z w.r.t.x...

    x[z_x] + y[z_y] + z = 0 (*)

    i have tried totally differentiating

    dz = {-[1/x^2][f(y/x)] + [1/x][f_x] }dx + [1/x][f_y]dy

    but then i get from the LHS of (*) that [1/x][f_x] }dx + [1/x][f_y]dy = ?

    I am given clues for two methods that i may follow:

    1) set y/x =v

    and

    2) to use the fact that for a homogenous function g(x,y) of nth order it is true that x[g_x] + y[g_y] = 0


    i am stuck
    The first hint looks interesting: v(x,y)=\frac{y}{x}\Longrightarrow z=\frac{f(v(x,y))}{x}\Longrightarrow z_x=\frac{x\,\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}-f(v)}{x^2} =\frac{-\frac{y}{x}\,\frac{\partial f}{\partial v}-f(v)}{x^2}\,,\,\,z_y=\frac{1}{x}\frac{\partial f}{\partial v}\frac{\partial v}{\partial y}=\frac{1}{x^2}\frac{\partial f}{\partial v}

    Well, now just verify that indeed xz_x+yz_y+z=0

    Tonio
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