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