D'Alembert, Lagrange, and Reduction of Order - The History

Our modern method for reduction of order is due to the French mathematician and physicist Jean le Rond d'Alembert (1717-1783) 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 $m-1$ 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 Joseph-Louis Lagrange (1736-1813).  To learn more, we examine the letters shared between these mathematicians.