Integrating Factor for First Order ODE

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 non-homogeneous and not exact.

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

is an exact differential equation.

Unfortunately, there is no systematic method of finding such a $\mu \left({x, y}\right)$ 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 $x y$

Technique for finding an Integrating Factor
Suppose, therefore, you were given a differential equation which is in (or can be manipulated into) the form:
 * $\displaystyle M \left({x, y}\right) + N \left({x, y}\right) \frac {\mathrm d y} {\mathrm d x} = 0$

and it was not homogeneous, exact or even linear.

Then what you can do is evaluate:
 * $\displaystyle \frac {\partial M}{\partial y} - \frac {\partial N}{\partial x}$

and see what you get when you divide it by each of $N$, $M$, $N - M$ and $N y - M x$ in turn.

Then examine what you get to see if you have a function in $x$ only, $y$ only, $x + y$ or $xy$ respectively.

If you do, then you have found an integrating factor and can solve the equation by using the technique defined in Solution to Exact Differential Equation.