Determine if each of the following functions is homogenous A

Determine if each of the following functions is

homogenous: A) x^2 - 3xy + y^2. B) x^2 + 3y - y^2. C) sqrt( 2x^4 + 6xy^3).

Enter (1) if homogeneous, or enter (0) if not homogeneous. ans:3

Solution

A Function f(x,y )is said to be Homogeneous Equation of degree n if

putting x=xt and y=yt we get f(xt,yt)= tⁿf(x,y)

[1] f(x,y) = x²-3xy+y² ; f(xt,yt) = (xt)²- 3(xt)(yt) + (yt)²=t²*( x²-3xy+y²)=t²* f(x,y)

Hence above equation is Homogeneous Equation of degree 2

[2]  f(x,y) = x²+3y-y² ; f(xt,yt) = (xt)² +3(yt) - (yt)²= x²t² + 3yt - y²t²

  which can not be written in  f(xt,yt)= tⁿf(x,y) form

  Hence above equation is NOT Homogeneous Equation

[3] f(x,y)= sqrt(2x⁴+6xy³) ;   f(xt,yt) =sqrt(2x⁴t⁴+6(xt)(yt)³)= sqrt(2x⁴t⁴+6xy³t⁴)=t²*sqrt(2x⁴+6xy³)=t² f(x,y)

  Hence above equation is Homogeneous Equation of degree 2