Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor

Proof Technique
The technique to solve a linear first order ordinary differential equation in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

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

is an integrating factor for $(1)$.

Multiplying it by:
 * $e^{\int \map P x \rd x}$

to reduce it to a form:
 * $\dfrac {\d y} {\d x} e^{\int \map P x \rd x} y = e^{\int \map P x \rd x} \map Q x$

is known as Solution by Integrating Factor.

It is remembered by the procedure:
 * Multiply by $e^{\int \map P x \rd x}$ and integrate.