Integrating Factor for First Order ODE

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 non-homogeneous and not exact.

By Existence of Integrating Factor, if $(1)$ has a general solution, there exists an integrating factor $\map \mu {x, y}$ such that:
 * $\displaystyle \map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \frac {\d y} {\d x} } = 0$

is an exact differential equation.

Unfortunately, there is no systematic method of finding such a $\map \mu {x, y}$ for all such equations $(1)$.

However, there are certain types of first order ODE for which an integrating factor can be found procedurally.

Proof
We have one of these:
 * Integrating Factor for First Order ODE: Function of One Variable: $x$ or $y$ only
 * Integrating Factor for First Order ODE: Function of $x + y$
 * Integrating Factor for First Order ODE: Function of $x y$