Question

Prove that f(x, y) = x³ – 2x²y + 3xy² + y³ is homogeneous. What is the degree? Verify Euler’s Theorem for f

Answer 1

f(x, y) = x³ – 2x²y + 3xy² + y³

f(λx, λy) = λ³x³ – 2λ²x²λy + 3λxλ²y² + λ³y³

= λ³(x³ – 2x²y + 3xy² + y³)

f is a homogeneous function of degree 3

By Euler’s Theorem,

`x  (del"f")/(delx) + y (del"f")/(dely) = 3"f"`

Verification:

f(x, y) = `x^3 - 2x^2y + xy^2 + y^3`

`(del"f")/(delx) = 3x^2 - 4xy + 3y^2`

`x (del"f")/(delx) = 3x^3 - 4x^2y + 3xy^2`

`(del"f")/(dely) = - 2x^2 6xy + 3y^2`

`y (delf)/(dely) = - 2x^2y + 6xy^2 + y^2`

`x (del"f")/(dely) + y (del"f")/(dely)` = 3x³ – 4x²y + 3xy² – 2x²y + 6xy² + 3y³

= 3x³ – 6x²y + 9xy² + 3y³

= 3(x³ – 2x²y + 3xy² + y³)

`x  (del"f")/(delx) + y (del"f")/(dely) = 3"f"`

We verified the Euler’s Theorem.