Question

A function f(x, y) is said to be homogeneous of degree n in the variables x and y if it can be expressed in the form
1. \({x^n}\phi \left( {\frac{y}{x}} \right)\)
2. \({y^n}\phi \left( {\frac{x}{y}} \right)\)
3. Both 1) and 2)
4. None of the above

Answer 1

Correct Answer - Option 3 : Both 1) and 2)

Concept:

Method 1.

To be called homogeneous a function \({\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right)\) must satisfy the below equation.

f (zx, zy) = zⁿ f (x, y)

n is the degree of homogeneity.

In other words, we need to follow the below steps to check for homogeneity:

1. Multiply each variable in the function f (x, y) with z to get f (zx, zy).
2. Then find if f (zx, zy) can be arranged to get zⁿ f (x, y) or not.
3. If we can get f (zx, zy) = zⁿ f (x, y) then homogeneous else not.

 

Method 2.

An expression of the form

f (x, y) = a₀xⁿ + a₁x^n-1 y + a₂ x^n-2 y² + … + a_nyⁿ

Where every term is of the nth degree is called a homogeneous function of degree n.

The equation can be rewritten as:

\({\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right) = {{\rm{x}}^{{\rm{n\;}}}}\left[ {{{\rm{a}}_0} + {{\rm{a}}_1}\left( {\frac{{\rm{y}}}{{\rm{x}}}} \right) + {{\rm{a}}_2}{{\left( {\frac{{\rm{y}}}{{\rm{x}}}} \right)}^2} + \ldots + {{\rm{a}}_{\rm{n}}}{{\left( {\frac{{\rm{y}}}{{\rm{x}}}} \right)}^{\rm{n}}}} \right]\)

\(\therefore {\bf{f}}\left( {{\bf{x}},{\bf{y}}} \right) = {{\bf{x}}^{{\bf{n}}\;}}\phi \left( {\frac{{\bf{y}}}{{\bf{x}}}} \right)\)

And also,

\({\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right) = {{\rm{y}}^{{\rm{n\;}}}}\left[ {{{\rm{a}}_0}{{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)}^{\rm{n}}} + {{\rm{a}}_1}{{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)}^{{\rm{n}} - 1}} + {{\rm{a}}_2}{{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)}^{{\rm{n}} - 2}} + \ldots + {{\rm{a}}_{\rm{n}}}} \right]\)

\(\therefore {\bf{f}}\left( {{\bf{x}},{\bf{y}}} \right) = {{\bf{y}}^{\bf{n}}}\phi \left( {\frac{{\bf{x}}}{{\bf{y}}}} \right)\)