Solution to Linear First Order Ordinary Differential Equation/Proof 1

Proof
Consider the first order ordinary differential equation:


 * $\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

We can put our equation:
 * $(1): \quad \dfrac {\d y} {\d x} + \map P x y = \map Q x$

into this format by identifying:
 * $\map M {x, y} \equiv \map P x y - \map Q x, \map N {x, y} \equiv 1$

We see that:
 * $\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} = \map P x$

and hence:
 * $\map P x = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x} } N$

is a function of $x$ only.

It immediately follows from Integrating Factor for First Order ODE that:
 * $e^{\int \map P x \rd x}$

is an integrating factor for $(1)$.

So, multiplying $(1)$ by this factor:
 * $e^{\int \map P x \rd x} \dfrac {\d y} {\d x} + e^{\int \map P x \rd x} \map P x y = e^{\int \map P x \rd x} \map Q x$

The result follows by an application of Solution to Exact Differential Equation.