Classification of Homogeneous functions

Question

Is every homogeneous function of degree 1 in two variables is of the form $f(x,y)=\frac{p(x,y)}{q(x,y)}$, where $p(x,y)$ is a homogeneous polynomial of degree $n$ and $ q(x,y)$ is a homogeneous polynomial of degree $n-1$. Or there are any other possibilities?

Actually I want to solve a non linear second order ODE and it is known that solution is homogeneous of degree $1$. The ODE is independent of $y$. Only $f$ and its derivatives with respect to $x$ is involve in the ODE. Can I use the above result? Or, I need to think in some other way.

Answer 1

Take any function $g: \mathbb R \to \mathbb R$. Then define $f(x,y)=xg(\frac y x)$ for $x \ne 0$, and $f(0,y)=0$.

$f$ is homogeneous of degree $1$ w.r.t. $(x,y)$. But some $f$ cannot be represented as quotients of polynomials, if only for a matter of dimension:

• the set of quotients of polynomials has dimension equal to the cardinal of $\mathbb N$ (each quotient is represented by two finite lists of coefficients)
• the set of all functions in $\mathbb R \to \mathbb R$ has a dimension at least that of the cardinal of $\mathbb R$ (it includes at least the linearly independent set of functions $\delta_r: \delta_r(r)=1, \delta_r(x\ne r)=0$, for any $r \in \mathbb R$).