14 votes
Accepted

Who invented the exponential ansatz for linear differential equations with constant coefficients?

The short answer is Euler. Some details are given in the following long quote from Hald's History of Probability and Statistics and Their Applications before 1750 (p.438): In 1743 Euler solved the ...
Conifold's user avatar
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12 votes
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Who did first use the Method of Characteristics?

[EDIT: check the (*) for a potential earlier reference in Lagrange] The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt ...
Laurent Duval's user avatar
10 votes
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Who wrote down the first differential equation?

It is impossible to tell "who wrote down" first. Leibniz PUBLISHED first (1684), but some documents indicate that he knew how to solve some differential equations in 1666. Newton never published his "...
Alexandre Eremenko's user avatar
8 votes
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History of hypergeometric equation

If $a=b=c=0$ we get a first order separable equation for $y'$, viz. $\ x[(1-x)y''-y']=0$ whose solution $y'= c(1-x)^{-1}$ would have been obvious since at least Bernoulli. Dutka (1984, p. 16): In ...
Francois Ziegler's user avatar
8 votes
Accepted

What is the origin of the term "Ordinary Differential Equation"?

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 ...
nwr's user avatar
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7 votes

source of "logistic growth"?

According to some on-line resources for historical French lexicon, around 1770 [see : Fortunato Bartolomeo De Felice, vol.26 of: Encyclopédie ou Dictionnaire universel raisonné des connoissances ...
Mauro ALLEGRANZA's user avatar
7 votes
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How did Peano prove his existence theorem without Ascoli's theorem?

Peano did do something else, and not quite right, apparently. In 1969 Kennedy published a note in the American Mathematical Monthly titled Is There an Elementary Proof of Peano's Existence Theorem, ...
Conifold's user avatar
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7 votes
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Source of claim that Leibniz discovered separation of variables for ODEs in 1691?

The reference is probably to a treatise sent to Huygens on 5 October 1691, where Leibniz says (and illustrates with several examples) that "Whenever the subtangent [$=y/y'$, but it would also work for ...
Viktor Blasjo's user avatar
7 votes

Does the “O” in the google doodle for Olga Ladyzhenskaya have anything to do with her work?

The equation in the picture is the Navier–Stokes momentum equation in convective form. The part inside the O looks like a cross-section of (the velocity field of) a flow between two rotating ...
Conifold's user avatar
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7 votes
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History behind the heat equation

Heat equation was first formulated by Fourier in a manuscript presented to Institut de France in 1807, followed by his book Theorie de la Propagation de la Chaleur dans les Solides the same year, see ...
Conifold's user avatar
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7 votes
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What is the origin of the "Japanese bracket"?

I can provide a partial response to this question. As noted in my earlier comment, the term "Japanese bracket" appears in the 1999 paper "Global existence of small solutions to the ...
Mark Yasuda's user avatar
  • 1,548
6 votes

History of hypergeometric equation

1) For $a=b=c=0$ we have equation $$ x(1-x) \dfrac {d^2y}{dx^2}-x\dfrac {dy}{dx}=0 $$ with solution $C_1 + C_2\log(1-x)$. This would probably not be considered hypergeometric, except in a degenerate ...
Gerald Edgar's user avatar
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6 votes
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D'Alembertian symbol $\Box$

The symbol seems due to Poincaré (1901, p. 456): For brevity we will introduce the following symbol, by setting, $$ \Delta\mathrm U -\mathrm K_0\frac{d^2\mathrm U}{dt^2}=\Box\mathrm U $$ Here $\...
Francois Ziegler's user avatar
6 votes
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What is the etymology of "phase space" of a dynamical system?

Here is a direct link to Nolte's Tangled Tale of Phase Space on Physics Today. Big takeaways: the name did not come from Liouville's oft-cited 1838 paper, and Boltzmann used "phase" without &...
Conifold's user avatar
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6 votes

Notations for Laplacian: $\nabla^2$ vs. $\Delta$

Jeff Miller's site gives the first occurrence of $\Delta$ as The symbol $\Delta$ for the Laplacian operator (also represented by $\nabla^2$) was introduced by Robert Murphy in 1833 in Elementary ...
nwr's user avatar
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5 votes
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Did Newton find the trajectory of a body moving in uniform gravity under the quadratic resistance law (the ballistic problem)?

No, he did not. Not in the first (1687), not in the second (1713), and not in the third (1726) edition, below I quote from Cajori's edition of Motte's translation of the latter. Since he was unable to ...
Conifold's user avatar
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5 votes

History of PDE's in the 19th Century

It belonged to both. For example, the most important work on PDE in 19th century was arguably Fourier's Analytic theory of heat. You don't have to read the book, to conclude that this was applied ...
Alexandre Eremenko's user avatar
5 votes

Earliest Instances of a Slope/Direction Field for a First-Order ODE

Johann Bernoulli explains the idea of a direction field quite explicitly (Modus generalis construendi omnes aequationes differentiales primi gradus, Acta Eruditorum, November 1694). He focusses on ...
Viktor Blasjo's user avatar
5 votes

What does the Fourier transform have to do with heat?

Not to sway others from providing a better answer from more interesting sources, but I think one clue can be found in the current version of the Wikipedia article for Joseph Fourier himself: There ...
natevw's user avatar
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5 votes
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Notations for Laplacian: $\nabla^2$ vs. $\Delta$

What follows is from A History of Vector Analysis by Michael John Crowe (1967; 1985 Dover corrected reprint). (from middle of p. 167) In Heaviside's later papers of 1883 and 1884 use was made of ...
Dave L Renfro's user avatar
5 votes
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Who was the first person in history to calculate the limit $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$?

First of all, what does it exactly mean to "calculate" this limit? The limit equals to $e$, but this is the definition of $e$. So one only has to prove that the limit exists, and then denote ...
Alexandre Eremenko's user avatar
5 votes

Who was the first person in history to calculate the limit $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$?

I followed up on a pointer given by @nwr in comments that Jacob Bernoulli first encountered $e$ in the context of computing continuous compound interest. Sure enough, the article "The number $e$&...
njuffa's user avatar
  • 6,516
5 votes
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Are adjoint operators historically related to integrating factors?

Yes, Birkhoff and Rota make an unnecessarily cryptic allusion to Lagrange's memoir Solution de différents problèmes de calcul intégral (1766), where he multiplies a linear equation by a factor and ...
Conifold's user avatar
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4 votes

What is the origin of the method of undetermined coefficients?

The method of undetermined coefficients was probably first developed by Euler, he applied it in his work to analyze the perturbations of Saturn by Jupiter for the Paris Academy's prize competition of ...
terry-s's user avatar
  • 4,445
3 votes

What were the early uses of differential equations for modeling chemical reactions?

An early result in chemical kinetics was Waage and Guldberg's law of mass action, that relates the speed of a chemical reaction to the quantity of the reacting substances. The field came of age in ...
Conifold's user avatar
  • 76k
3 votes

source of "logistic growth"?

As Mauro states, the term logistic is due to the Belgian mathematician Pierre François Verhulst, who invented the logistic growth model, and named it logistic (French: logistique) in his 1845 "...
Nils von Barth's user avatar
3 votes
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Origin of the terminology “trace operator” related to boundary-value problems for PDEs

Normally, I'd refer to Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics site for these kinds of questions, but it is silent on this one, and generally "trace"'s history seems ...
Conifold's user avatar
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3 votes
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Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?

A paper of Kneser (1904) strongly suggests that the idea does (indeed) go back to Cauchy, in connection with Sturm-Liouville problems (i.e. ordinary differential operators, as opposed to the Laplacian ...
Francois Ziegler's user avatar
3 votes

Who wrote down the first differential equation?

"But this will appear plainer by an Example or two. ..." (Newton 1671) --- After outlining his general method for finding solutions of differential equations. and Newton obtained the solution of ...
Gerald Edgar's user avatar
  • 10.3k
3 votes

Poincaré's last geometric theorem

Poincaré was studying the circular restricted three-body problem, where two bodies in a binary system revolve around their barycenter and a third test mass moves in some orbit around them. Birkhoff ...
HDE 226868's user avatar
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