Which one of the following function(s) is/are homogeneous? `(a) f(x,y)= (x-y)/(x^2 + y^2)` (b) `f(x,y)= x^(1/3)y^(-2/3)tan^-1(x/y)` (c) `f(x,y)=x(lnsqrt(x^2+y^2)-lny)+ye^(x/y)` (d) none of these

Options

A. `f(x,y) = (x-y)/(x^(2)+y^(2))`
B. `f(x,y) = x^(1/3)y^(-2/3) tan^(-1)x/y`
C. `f(x,y) = x(" ln "sqrtx^(2)+y^(2))-" ln "y+ye^(x//y)`
D. `f(x,y) = x[" ln "(2x^(2)+y^(2))x -" ln "(x+y)]+y^(2)tan(x+2y)/(3x-y)`

Answer

A, B, C

Explanation

a) `f(lambdax,lambday)=(lambda(x-y))/(lambda^(2)(x^(2)+y^(2)))=lambda^(-1)f(x,y)`
Thus, it is homogenous of degree `-1`
b) `f(lambdax,lambday)=(lambdax)^(1//3)(lambday)^(-2//3)tan^(-1)x/y`
`=lambda^(-1//3)x^(1//3)tan^(-1)x/y`
`=lambda^(-1//3)f(x,y)`
c) `f(lambdax,lambday) = lambdax("ln"sqrt(lambda^(2)(x^(2)+y^(2))-"ln "lambday))+lambdaye^(x//y)`
`=lambdax["ln"((lambdasqrt(x^(2)+y^(2)))/(lambday))]+lambdaye^(x//y)`
`lambda[x("ln "sqrt(x^(2)+y^(2))-"ln"y)+ye^(x//y)]`
`=lambdaf(x,y)`
Thus, it is homogeneous.
d) `f(lambdax,lambday)=lambdax["ln "(2lambda^(2)x^(2)+lambda^(2)y^(2))(lambdaxlambda(x+y))]+lambda^(2)x^(2)tan(x+2y)/(3x-y)`
`=lambda x["ln "(2x^(2)+y^(2))/(x(x+y))]+lambda^(2)x^(2)tan(x+2y)/(3x-y)`
Thus, it is non-homogeneous.