DETERMINE IF a FUNCTION IS HOMOGENOUS AND FIND DEGREE

Let A = {(x, y) a < x < b, c < y < d} ∈ 2, F : A-> ℝ, we say that F is a homogeneous function on A, if there exists a constant P such that F(λx, λy) = λp f(x, y) for all λℝ such that (λx, λy)∈A. This constant is called degree of F. 

Step 1 :

In the given function, apply x = λx and y = λy.

Step 2 :

Do the possible simplification.

Step 3 :

Get the function in the form of λp f(x).

P is the degree of the polynomial.

Problem 1 :

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.

(i) f (x, y) = x2 y + 6x3+7

Solution :

Apply x = λx and y = λy

f (λx, λy) = (λx)2 (λy) + 6(λx)3+7

f (λx, λy) = λ2xy + 6λ3x3+7

We cannot factor λ from the function. So it is not a homogeneous function.

(ii)

Solution :

=  λ3(6x2y3πy5+9x4y)/(2020x2+2019y2)

It is a homogenous function and its degree is 3.

(iii) 

Solution :

λ0√(3x2+5y2+z2)/(4x+7y)

So, it is a homogenous function of degree 0.

(iv)

Solution :

It is not a homogenous function.

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