Quotient of Homogeneous Functions
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Theorem
Let $\map M {x, y}$ and $\map N {x, y}$ be homogeneous functions of the same degree.
Then:
- $\dfrac {\map M {x, y} } {\map N {x, y} }$
is homogeneous of zero degree.
Proof
Let:
- $\map Q {x, y} = \dfrac {\map M {x, y} } {\map N {x, y} }$
where $M$ and $N$ are homogeneous functions of degree $n$.
Let $t \in \R$.
Then:
\(\ds \map Q {t x, t y}\) | \(=\) | \(\ds \frac {\map M {t x, t y} } {\map N {t x, t y} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {t^n \map M {x, y} } {t^n \map N {x, y} }\) | as $M$ and $N$ are homogeneous of degree $n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds t^0 \frac {\map M {x, y} } {\map N {x, y} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds t^0 \map Q {x, y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map Q {x, y}\) |
The result follows from the definition.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.7$: Homogeneous Equations