Who has first used the term "ordinary differential equation"? Is it known, why the term "ordinary" is used here? What makes an ODE "ordinary"?
The name "Ordinary Differential Equation", together with an explanation of why the term "ordinary" is used, is found in 1828 in An Elementary Treatise on the Differential and Integral Calculus by Jean-Louis Boucharlat and Ralph Blakelock:
If x, instead of being a function of two variables x and y, should contain only x, this would be no more than an ordinary differential equation, which, being integrated, would give...
It is interesting to note that the name "Partial Differential Equation" appears to predate the first use of ODE by almost 60 years. From the same site, we have the first use of "Partial Differential Equation" given as :
Partial differential equation was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in the title "Memoire sur les Equations aux différence partielles," which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCL&ldquo:III (1773).
The first English use is given as :
Partial differential equation appears in English in 1809 in a letter from “Mr. Thomas Knight, Of Papcastle, near Cockermouth” in The Mathematical Repository, New Series, Volume III (1809). The same issue of The Mathematical Repository contains the expression partial fluxion. [James A. Landau]