Existence of Infinitely Many Integrating Factors

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