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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 homogeneous linear differential equation of the m-th order with constant coefficients (using the same idea as de Moivre) by guessing at a particular ...

8

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 contrast to the geometric progression $a, ar, ar^2,\dots$ in which each term, after the first, is obtained by multiplying the preceding term by a constant ...

7

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 humaines (1773)] : [Adjectivement] Logarithmes logistiques, logarithmes dans lesquels zéro est le logarithme correspondant au nombre 3600. Ces logarithmes sont ...

7

The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt (?-1784). According to I. Grattan-Guinness et S. Engelsman, The manuscripts of Paul Charpit, Historia Mathematica, vol. 9, 1982, p. 65-75: He presented his paper on partial differential equations to the Académie on June 30,...

7

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 just the tangent $y'$] is a product of two quantities or formulas, of which one is given purely in terms of the abscissa $x$, and the other in terms of the ...

6

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 cylinders, mentioned in Ladyzhenskaya's Mathematical Analysis of Navier-Stokes Equations for Incompressible Liquids, 6.4, among other places, the so-called Taylor-...

6

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 sense. 2) Answer (needs reference): In a geometric series $\sum a_n$, the ratio $\frac{a_{n+1}}{a_n}$ is constant, does not depend on $n$. We generailze that ...

6

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, where he wrote: "In 1886 Giuseppe Peano stated [8] that the initial value problem: $y' = f(x,y),\, y(a) = b$, has a solution on the sole condition that $f$ is ...

6

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 $\mathrm K_0=1\,/\,c^2$ (p. 336). It didn’t immediately catch on: it appears without comment in Poincaré’s (1906, p. 132) (cited by Wikipedia), but not in the ...

5

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 find the trajectory analytically he tried to approximate it by pre-chosen curves. In the process he apparently messed up with the semi-circles in the first ...

4

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 drawing isoclines rather than slope segments. There is no figure in that work but Bernoulli drew an example in his correspondence: Corresponding to: Dominique ...

3

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 1884 with van'tHoff's "Etudes de dynamique chimique", which is filled with differential equations (see the English 1895 translation), he was awarded the first ...

3

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 "Recherches mathématiques sur la loi d'accroissement de la population", p. 8: Nous donnerons le nom de logistique à la courbe We will give the name ...

3

Poincare 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 acknowledges this in his proof of the theorem, Proof of Poincare's Geometric Theorem. Specifically, Poincare was looking to prove that this third body may have ...

3

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 unexplored. The first use of "trace" in the sense of restriction to the boundary on MathSciNet is (as far as I could find) in the review of Slobodeckii's 1958 note ...

3

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 in the body of your question). Given functions $g,k,l$ and writing $\smash{L=\frac d{dx}\left(k\frac{d}{dx}\,\cdot\right) - l},$ Kneser considers the [for us: “...

3

A. G. Greenhill, The applications of elliptic functions, Macmillan, London & NY, 1892, pp. 87-88. If you read French, a much clearer and more comprehensive discussion is in G.-H. Halphen, Traite de fonctions elliptiques et de leurs applications, 2-eme partie, Paris Gauthier-Villars et fils, 1888. Chapitre V.

2

"Indicial" is derived from "indices", in this case indices of the coefficients. The Latin word indicium originally meant a sign, indicator. "Indicial equation" is most often used in the context of solving equations of the form $a_nx^n\frac{d^n}{dx^n}+\dots+a_1x\frac{dy}{dx}+a_0y=0$. A substitution, known already to Euler, reduces it to a linear equation with ...

2

Side remark: Huygens had previously used that name for the graph of logarithm, in Discours de la cause de la pesanteur (1690, p. 169): ... there was a curved line, which I had examined much before, which was of great use in this research. One can call it Logarithmic or Logistic, for I don’t see that anyone has yet named it, although others also considered ...

2

According to Whittaker this problem was reduced to quadrature by D'Alembert in 1744: Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, p. 229 http://archive.org/details/treatisanalytdyn00whitrich Conifold's answer says Bernoulli solved it in a more general form in 1717. Presumably there is some difference between the types of ...

2

A celebrated textbook on enzyme kinetics is Fundamentals of Enzyme Kinetics by Athel Cornish-Bowden, Fourth Edition, 2012, Wiley-VCH, Weinheim. https://www.wiley-vch.de/home/fundenzykinet As a sample, one can download the first chapter for free, which intersperses historical references with a discussion of the basic principles of enzyme kinetics. ...

1

Adding to Bence Mélykúti's response, you can find a discussion of the relative contributions of Michaelis and Menten on the one hand and Henri on the other in a paper by Deichmann et al. "Commemorating the 1913 Michaelis-Menten paper Die Kinetik der Invertinwirkung: three perspectives" FEBS J. 281 (2014) 435-463. The "three perspectives" have different ...

1

This question involves two rather different points. It involves Newton's practices for notating not only derivatives, but also the functions (or whatever else) he used to indicate the values of those derivatives. In cases where we might now write dy/dx = {some-function} , the question involves Newton's practices for expressing both the left and the right ...

1

Sophus Lie's 1895 paper "Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung [General Theory of Partial Differential Equations of Arbitrary Order]" begins with a historical summary in which he seems to think Monge was the first (p. 4 of the English translation): …theory of characteristics [Theorie der Charakteristiken] ...

1

As a graphical method for the integration of hyperbolic PDEs the method of characteristics was proposed by the Belgian engineer Massau in the late 19th century. https://archive.org/stream/mmoiresurlintgr05massgoog#page/n4/mode/2up

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