Author(s):
Sarah Cummings (Wittenberg University) and Adam E. Parker (Wittenberg University)
Our modern method for reduction of order is due to the French mathematician and physicist Jean le Rond d'Alembert (17171783) in 1766 [3, p. 381]. Figure 1 shows d'Alembert published exactly our method.

Your problem on integrating the equation \(Py+\frac{Q dy}{dx}+\frac{Rd^2y}{dx^2} \dots + \frac{M d^my}{dx^m}=X\) when one has \(m1\) values of \(x\) [sic] in the equation \(Py+\frac{Q dy}{dx}+\frac{Rd^2y}{dx^2} \dots + \frac{M d^my}{dx^m}=0\) seems so beautiful to me that I've looked for a solution myself, which follows.
Let \(y=V\xi\), \(V\) being undetermined, and \(\xi\) one of the values of \(y\) that satisfies the equation \(Py+\frac{Q dy}{dx}+ \dots \& c.=0\) and so this value be substituted in the equation \(Py+\frac{Q dy}{dx}+ \& c. \dots=X\). The transformed will be composed of, 1) one part \(V(P\xi+\frac{Q d\xi}{dx}\dots + \frac{M d^m\xi}{dx^m})\)… where \(X\) does not exist, so it will evidently \(=0\), because \(P\xi +\frac{Q d\xi}{dx}\dots + \frac{M d^m\xi}{dx^m}=0\)...

Figure 1. D'Alembert explained what remains the modern method for reducing the order of a linear differential equation in 1766 [3, p. 381]. D'Alembert's French is followed by the authors' translation into English.
What is interesting, and what provides motivation for the rest of this paper is the title of his article: “Extrait de différentes lettres de M. d'Alembert à M. de la Grange écrites pendant les années 1764 & 1765" (“Excerpt from different letters between Mr. d'Alembert and Mr. de LaGrange written during the years 1764 & 1765”). It appears that this technique comes from a conversation with JosephLouis Lagrange (17361813). To learn more, we examine the letters shared between these mathematicians.
Sarah Cummings (Wittenberg University) and Adam E. Parker (Wittenberg University), "D'Alembert, Lagrange, and Reduction of Order  The History," Convergence (September 2015)