Question

$F(x,y)=x^3+y^3+3x^{2}y+3x{y}^2$. Is this homogeneous function.

Answer 1

$F(x,y)$ is a homogeneous function of degree $n$ if, for any real number $t$ ,

$F(tx,ty)=t^{n}F(x,y)$

Here, $F(x,y)=x^3+y^3+3x^{2}y+3x{y}^2$

$F(tx,ty)=(tx{)}^3+(ty{)}^3+3(tx{)}^{2}ty+3(tx\left)\right(ty{)}^2$

$=t^3(x^3+y^3+3x^{2}y+3x{y}^2)$

$=t^3\left(F\right(x,y\left)\right)$

So, $F(x,y)$ is a homogeneous function of degree $3$