Definition:Differential Equation/Historical Note
Historical Note on Differential Equations
According to H.T.H. Piaggio, the first person to solve a differential equation was Isaac Newton, which he did in $1676$ by use of an infinite series, $11$ years after he had invented the differential calculus in $1665$.
These results were not published till $1693$, the same year in which a differential equation occurred in the work of Gottfried Wilhelm von Leibniz, whose own work on differential calculus was published in $1684$.
However, E.L. Ince states that the term differential equation was first used by Gottfried Wilhelm von Leibniz (as æquatio differentialis) also in $1676$, to denote a relationship between the differentials $\d x$ and $\d y$ of two variables $x$ and $y$.
Jacob Bernoulli and Johann Bernoulli reduced a large number of differential equations into forms that could be solved.
Much of the theory of differential equations was established by Leonhard Paul Euler.
Joseph Louis Lagrange gave a geometrical interpretation in $1774$.
The first existence proof for the solutions of a differential equation was provided by Augustin Louis Cauchy.
He proved in $1823$ that the infinite series obtained from a differential equation is convergent.
The theory in its present form was not presented until the work of Arthur Cayley in $1872$.
Piaggio references the $1888$ work of Micaiah John Muller Hill.
Cauchy's work was continued by Charles Auguste Briot and Jean-Claude Bouquet
The Method of Successive Approximations was introduced by Charles Émile Picard in $1890$.
Lazarus Immanuel Fuchs and Ferdinand Georg Frobenius investigated linear differential equations of second order and higher with variable coefficients.
Marius Sophus Lie contributed his Lie's Theory of Continuous Groups revealed a connection between techniques which had previously been believed to be disconnected.
Graphical considerations were developed by Karl Hermann Amandus Schwarz, Felix Klein and Édouard Jean-Baptiste Goursat.
Takeo Wada extended these methods to the results of Charles Émile Picard and Jules Henri Poincaré.
Numerical methods were developed by Carl David Tolmé Runge, among others.
Sources
- 1926: E.L. Ince: Ordinary Differential Equations ... (next): Chapter $\text I$: Introductory: $\S 1.1$ Definitions
- 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Historical Introduction
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($\text {1789}$ – $\text {1857}$)