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Question
In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.
U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`
Solution
U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`
`U"(lambdax, lambday, lambdaz) = lambdax lambday + sin((lambda^2y^2 - 2lambda^2z^2)/(lambdaxlambday))`
= `lambda^2xy + sin((lambda^2(y^2 - 2z^2))/(lambda^2(xy)))`
= `lambda^2xy + sin ((y^2 - 2z^2)/(xy))`
There is no common λ
∴ It is not homogeneous.
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