Editing Differential equation (section)

==History==

Differential equations came into existence with the [[History of calculus|invention of calculus]] by [[Isaac Newton]] and [[Gottfried Leibniz]]. In Chapter 2 of his 1671 work [[Method of Fluxions|''Methodus fluxionum et Serierum Infinitarum'']],<ref>Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].</ref> Newton listed three kinds of differential equations:

:<math>\begin{align}
  \frac {dy}{dx} &= f(x) \\[4pt]
  \frac {dy}{dx} &= f(x, y) \\[4pt]
  x_1 \frac {\partial y}{\partial x_1} &+ x_2 \frac {\partial y}{\partial x_2} = y
\end{align}</math>
In all these cases, {{mvar|y}} is an unknown function of {{mvar|x}} (or of {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}), and {{mvar|f}} is a given function.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

[[Jacob Bernoulli]] proposed the [[Bernoulli differential equation]] in 1695.<ref>{{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis | year=1695 | journal=[[Acta Eruditorum]]}}</ref> This is an [[ordinary differential equation]] of the form

: <math>y'+ P(x)y = Q(x)y^n\,</math>

for which the following year Leibniz obtained solutions by simplifying it.<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>

Historically, the problem of a vibrating string such as that of a [[musical instrument]] was studied by [[Jean le Rond d'Alembert]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph-Louis Lagrange]].<ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin of the American Mathematical Society |series=New Series |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first1=Gerard F. |last1=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] {{Webarchive|url=https://web.archive.org/web/20200209023122/http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= |date=2020-02-09 }} (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> In 1746, d’Alembert discovered the one-dimensional [[wave equation]], and within ten years Euler discovered the three-dimensional wave equation.<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref>

The [[Euler–Lagrange equation]] was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]].

In 1822, [[Joseph Fourier|Fourier]] published his work on [[heat flow]] in ''Théorie analytique de la chaleur'' (The Analytic Theory of Heat),<ref>{{Cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = fr | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref> in which he based his reasoning on [[Newton's law of cooling]], namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his [[heat equation]] for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.