Integrating Factor for First Order ODE/Function of Sum of Variables

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.

Suppose that:
 * $\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } {\map N {x, y} - \map M {x, y} }$

is a function of $z$, where $z = x + y$.

Then:
 * $\map \mu {x + y} = \map \mu z = e^{\int \map g z \rd z}$

is an integrating factor for $(1)$.

Proof for Function of $x + y$
Suppose that $\mu$ is a function of $z = x + y$.

Then:
 * $\dfrac {\partial z} {\partial x} = 1 = \dfrac {\partial z} {\partial y}$

Thus:
 * $\dfrac {\partial \mu} {\partial x} = \dfrac {d \mu} {d z} \dfrac {\partial z} {\partial x} = \dfrac {\d \mu} {\d z} = \dfrac {\d \mu} {\d z} \dfrac {\partial z} {\partial y} = \dfrac {\partial \mu} {\partial y}$

which, when substituting in $(3)$, leads us to:
 * $\dfrac 1 \mu \dfrac {\d \mu} {\d z} = \dfrac {\map P {x, y} } {\map N {x, y} - \map M {x, y} } = \map g z$

where $\map g z$ is the function of $z$ that we posited.