# 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 ($\text {1789}$ – $\text {1857}$)