Thread: 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

Originally Posted by pepsi

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:

Well, now just verify that indeed

Tonio