How do you determine if #f(x,y)=(xy)/sqrt(x^2+y^2)# is homogeneous and what would it's degree be?

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Answer 1 · 2017-02-19T20:09:44

#f(x)# is homogeneous of degree #1#

We have to evaluate:

#f(alpha x, alpha y) = ((alpha x) (alpha y))/sqrt((alpha x)^2+(alpha y)^2)#

#f(alpha x, alpha y) = (alpha^2xy)/sqrt(alpha^2 (x^2+y^2))#

#f(alpha x, alpha y) = (alpha^2xy)/(alpha sqrt (x^2+y^2))#

#f(alpha x, alpha y) = (alphaxy)/sqrt (x^2+y^2)#

So we have:

#f(alpha x, alpha y) = alpha f(x,y)#

and we can conclude that #f(x)# is homogeneous of degree #1#.