Quotient of Homogeneous Functions

Theorem

Let $M \left({x, y}\right)$ and $N \left({x, y}\right)$ be homogeneous functions of the same degree.

Then:

$\dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

is homogeneous of degree zero.

Proof

Let:

$Q \left({x, y}\right) = \dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

where $M$ and $N$ are homogeneous functions of degree $n$.

Let $t \in \R$. Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle Q \left({tx, ty}\right)\)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac {M \left({tx, ty}\right)} {N \left({tx, ty}\right)}\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac {t^n M \left({x, y}\right)} {t^n N \left({x, y}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	as $M$ and $N$ are homogeneous of degree $n$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle t^0 \frac {M \left({x, y}\right)} {N \left({x, y}\right)}\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle t^0 Q \left({x, y}\right)\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle Q \left({x, y}\right)\)	\(\displaystyle \)	\(\displaystyle \)

The result follows from the definition.

$\blacksquare$

Sources

• George F. Simmons: Differential Equations (1972): $\S 7$