Who's the "Author" of the integrating factor method? I've always been interested in how someone discovered this method, it felt pretty magical when I first learnt it, and I've been wondering who discovered/how was it derived for the first time.
Does anyone here happen to know this?
 A: The integrating factor is referred to as Euler's multiplier in honour of Euler for using it to reduce second order equations. See J. Sasser. “History of Ordinary Differential Equations: the First Hundred Years”,  
He used as the base $c$ not  the natural logarithm $e$ as is used today.
But the notion of multiplying by an "integrating factor" was also used earlier by  Bernouli,  using $$\frac{x}{y^2}$$  to reduce the first order equation $$2ydx-xdy = 0$$ to $$\frac{d}{dx}(\frac{x^2}{y}) = 0$$  See Mathematical Time Capsules, MAA, page 212
EDIT:
As suggested in comment by user @A.G., it has its roots with Leibniz as well.
By 1694, Leibniz solved the general first order equation $m dx + n y dx + dy= 0$, where $m, n$ are functions of $x$, by introducing $$\frac{1}{p}dp = n dx$$ which is in modern terms $p = e^{\int n dx}$. Now, the ODE becomes $p m dx + (y p)' = 0$, i.e. $$y = -\frac{\int p(x) m(x) dx}{p}$$
See Chapter 17, A History of Mathematics, Victor Katz.
A: About the name of the method:

INTEGRATING FACTOR is found in May 1845 in a paper by Sir George Gabriel Stokes published in the Cambridge Mathematical Journal [University of Michigan Historical Math Collection]. 

(Source)
About the development of the method:

This capsule offers some differential equations solved by the
  originators of the technique of using an integrating factor, though
  they did not use that expression. (..) Memorizing the formula would
  not be in the spirit of the originators of the method, Johann
  Bernoulli (1667–1748) and Leonhard Euler (1707–1783), (..)

(Source)
