A function f(x, y) is called homogeneous of degree n if it satisfies the equation f(tx, ty) = t n f(x, y) for all t, where n is a positive integer. Show that if f(x, y) is homogeneous of degree n, then x ∂ f ∂ x +y ∂ f ∂ y = n f(x, y). (Hint: Try differentiating: find ∂ ∂t t=1 of a homogeneous function f in two ways.)

It is proved that .

Step-by-step explanation:

Given function is,

for all t>0 and n is positive integers.

To show,

Let, u=tx and v=ty, then,

differentiate partially with respect to x we get,

[tex}\frac{\partial}{\partial x}f(u,v)=\frac{\partial}{\partial x}(t^nf(x, y))[/tex]

Again, differentiate (1) with respect to t we get,

      (By putting values of )

Hence proved.

equal to

step-by-step explanation: