Existence of Infinitely Many Integrating Factors
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Theorem
Let the first order ordinary differential equation:
- $(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be such that $M$ and $N$ are real functions of two variables which are not homogeneous functions of the same degree.
Suppose $(1)$ has an integrating factor.
Then $(1)$ has an infinite number of integrating factors
Proof
Let $\map F f$ be any function of $f$ which is an integrating factor of $(1)$.
Then:
- $\ds \mu \map F f \paren {\map M {x, y} \rd x + \map N {x, y} \rd y} = \map F f \rd f = \map \d {\int \map F f \rd f}$
so $\mu \map F f$ is also an integrating factor.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.9$: Integrating Factors