Suppose a function f x y satisfies the identity f tx ty tnf

Suppose a function f (x, y) satisfies the identity f (tx, ty) = tⁿ*f (x, y) for some
fixed n. Show that
x*(f/x)+ y*(f/y)= n*f (x, y).

Solution

Answer:

Suppose a function f (x, y) satisfies the identity f (tx, ty) = tⁿ*f (x, y) for some fixed n.

differentiate with respect to t , we get by chain rule

x*(f/x)+ y*(f/y)= nt^(n-1)*f (x, y).

for t=1 we get x*(f/x)+ y*(f/y)= n*f (x, y).

hence the result is proved