Existence of Infinitely Many Integrating Factors

Theorem
Let the first order ordinary differential equation:
 * $(1): \quad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

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 $F \left({f}\right)$ be any function of $f$.

Then:
 * $\displaystyle \mu F \left({f}\right) \left({M \left({x, y}\right) \, \mathrm d x + N \left({x, y}\right) \, \mathrm d y}\right) = F \left({f}\right) \, \mathrm d f = \mathrm d \left({\int F \left({f}\right) \, \mathrm d f}\right)$

so $\mu F \left({f}\right)$ is also an integrating factor.