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:
 * $\dfrac {\mathrm d y}{\mathrm d x} + P \left({x}\right) y = Q \left({x}\right)$

It immediately follows from Integrating Factor for First Order ODE that:
 * $e^{\int P \left({x}\right) dx}$

is an integrating factor for $(1)$.

Multiplying it by:
 * $e^{\int P \left({x}\right) \, \mathrm d x}$

to reduce it to a form:
 * $\dfrac {\mathrm d y}{\mathrm d x} e^{\int P \left({x}\right) \, \mathrm d x} y = e^{\int P \left({x}\right) \, \mathrm d x} Q \left({x}\right)$

is known as Solution by Integrating Factor.

It is remembered by the procedure:
 * Multiply by $e^{\int P \left({x}\right) \, \mathrm d x}$ and integrate.