Types of solutions of differential equations

Part of the Science Networks. Historical Studies book series (SNHS, volume 35)


This chapter deals with several aspects of differential equations relating to types of solutions (complete, general, particular, and singular integrals or solutions), as opposed to methods of solution. That is, the subject here is not so much the processes for solving differential equations, as the conceptions about what kind of object a final solution might be. For this reason, the word “solution” will be used here in the sense of answer, but not in the sense of process for obtaining an answer.


Arbitrary Function Arbitrary Constant Eighteenth Century Geometrical Interpretation Singular Integral 
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  1. 5.
    He probably used this adjective because Taylor [1715], when encountering for the first time a singular solution, had remarked that it was “singularis quædam solutio”, which may be translated as “a certain unique solution” — “unique” either in the sense of only one (of its kind) or of remarkable.Google Scholar
  2. 6.
    But not immediately: in the 1820’s the syllabi of the École Polytechnique still used Laplace’s term “particular solutions” [Gilain 1989, 112, 116, 120, 126, 130], while Cauchy, in his lectures there, changed from following that in 1819/1820 and 1821/1822 [Gilain 1989, 61, 67] to speaking of “singular integrals” in 1823/1824, 1827/1828 and 1829/1830 [Gilain 1989, 73, 85, 93].Google Scholar
  3. 7.
    Except for authors (possibly influenced by Laplace [1772a]) who seemed to prefer “general integral” as the principal term: the syllabi of the École Polytechnique from 1817 to 1830 spoke of “general integrals” of ordinary differential equations [Gilain 1989, 108–130], and so did Cauchy in his lectures [Gilain 1989, 56–94]. But among the authors studied here Laplace and Condorcet [1765, 3, 67] were the only ones with that preference.Google Scholar
  4. 8.
    Brook Taylor had encountered one before that, but he does not seem to have noticed its significance [Taylor 1715, 26–27].Google Scholar
  5. 9.
    x = a for \( ds = \frac{{ - dx\sqrt b }} {{\sqrt {a^2 b - bx^2 - a^2 {\text{ }}\smallint P{\text{ }}dx} }} \) and k2u = (k2 + 1)x for \( \frac{{(k^2 + 1)dx - k^2 du}} {{\sqrt {(k^2 + 1)x - k^2 u} }} = \frac{{ \pm du}} {{\sqrt u }} \) [Euler 1736, § 335, § 300].Google Scholar
  6. 11.
    [Blanc 1957, xx] presumes that some real mistake had slipped into Euler’s reasonings. He is very critical of the whole memoir [Euler 1764].Google Scholar
  7. 15.
    As long as certain conditions apply: that at least one of x, y appears in dy/da = 0 [Lagrange 1774, § 4] and that not all of dy/da, d2y/dxda, d3y/dx2da,... are zero (see below).Google Scholar
  8. 19.
    Before [Lagrange 1774] “general integral” had been simply an alternative name for “complete integral”: we have seen above Laplace using it in that sense.Google Scholar
  9. 20.
    It is true that in [Euler Integralis, II] (published in 1769) there are two chapters which refer to construction of ordinary differential equations: chapters 10 and 11 of the first section, respectively “de constructione aequationum differentio-differentialium per quadraturas curvarum” (“on the construction of differentio-differential [i.e., second-order differential] equations by quadratures of curves”) and “de constructione aequationum differentio-differentialium ex earum resolutione per series infinitas petita” (“on the construction of differentio-differential [i.e., second-order differential] equations from their required solution by infinite series”). But one would seek in vain for geometrical constructions in those chapters. Rather, Euler seems to refine problems and techniques which had appeared in the context of construction of differential equations (namely solving an equation assuming certain quadratures or rectifications — see below), but which appear devoid of geometrical meaning. Deakin [1985] finds integral transforms in chapter 10.Google Scholar
  10. 33.
    With one possible exception, striking but very isolated: according to Youschkevitch [1976, 71] (following Truesdell) Euler [1765c, § 39] introduced pulse functions (different from zero only at one point); Lützen [1982, 197–198] disagrees with the implication in Youschkevitch’s text that those were delta functions; for myself, I am not completely convinced that Euler thought he was talking about functions at all.Google Scholar
  11. 34.
    That is, d’Alembert effectively accompanied Euler in this evolution: in [d’Alembert 1747], the concept of function is the same as in [Euler Introductio]; while [d’Alembert 1780] is a memoir on discontinuous functions.Google Scholar
  12. 43.
    Condorcet [1771, 69–71] and Laplace [1779, 299–302] imposed a less strict condition: the functions and the derivatives up to order n − 1 were forbidden to have “jumps”, but the n-th derivative was not. Note however that Laplace admitted stronger discontinuities in physical (rather than “geometrical”) solutions, using an argument similar to Euler’s number 3 below [1779, 302]. See also section 9.5.4.Google Scholar
  13. 46.
    Lützen [1982, 19] sees this as an anticipation of the most common technique in the 20th century for obtaining generalized solutions to differential equations, although this technique consists in replacing the differential equation with different types of integral equations.Google Scholar
  14. 51.
    Euler even found himself a serious objection to his geometrical correspondence between curves and arbitrary functions: integrating \( (\frac{{ddz}} {{dy^2 }}) + aa(\frac{{ddz}} {{dx^2 }}) = 0 \) he arrived at z = \( z = f:(x + ay\sqrt { - 1} ) + F: (x - ay\sqrt { - 1} ) \) ; what could an abscissa like \( x + ay\sqrt { - 1} \) mean, not even he had any idea [Euler Integralis, III, § 301; Ferraro 2000, 128–129]. Nevertheless, Ferraro [2000, 130] exaggerates when he says that the objects that Euler called discontinuous functions “substantially differed from effective functions since only the latter could be manipulated and, therefore, accepted as solutions to a problem”; the vibrating-string controversy shows that Euler did accept discontinuous functions as solutions, and strived to be able to manipulate them.Google Scholar
  15. 53.
    In a letter to Condorcet dated 2nd September 1771 (published by Taton [1947, 979–982]), he had given all of these equations plus \( \frac{{x\delta z}} {{dx}} + \frac{{y\partial z}} {{dy}} = z - a \) for a conical surface.Google Scholar
  16. 55.
    Not so for the general 18th-century reader who did not know the manuscript of [Monge 1771].Google Scholar
  17. 65.
    Lagrange assumes that these lines must intersect consecutively (or be parallel, which may be interpreted as intersection at infinity). In the case of straight lines this makes him miss the case of skew surfaces [Lacroix Traité, I, 501].Google Scholar
  18. 66.
    The fifth and final “article” in [Lagrange 1779] is also very different, but in another sense: it is there that Lagrange presents his method for integrating quasi-linear first-order partial differential equations. The connection with singular integrals is that it is a generalization of a method given in [Lagrange 1774, § 52]. A geometrical example is given, but it is irrelevant for us here.Google Scholar
  19. 67.
    We cannot exclude the possibility that Monge was inspired in this by the fourth article of [Lagrange 1779], which appeared in 1781, but apropos of a completely different issue Monge claimed later not to have known [Lagrange 1779] (in [Monge 1784b, 118], which according to Taton [1951, 289] was submitted only in 1786). The issue there was Lagrange’s method for integrating quasi-linear first-order partial differential equations, which appeared in the fifth article of [Lagrange 1779].Google Scholar
  20. 70.
    In fact, a slightly different version of this condition had already been found by Fontaine [Greenberg 1982, 12, 20–26]. Clairaut, although critical of Fontaine’s style, acknowledged his priority [1740, 310]. Furthermore, Cousin [1796, I, 258] attributed (6.13) to “N. Bernoulli”-presumably Nicolaus (I) Bernoulli, in an extract of a letter published in an article by his cousin Nicolaus (II) Bernoulli [1720, 442–443] (see [Engelsman 1984, 186–187] for the unravelling of this “bibliographical monster”, which had been cited by Poggendorff and Fleckenstein as it if were an independent article, with a wrong date, and in the latter case with wrong page numbers — and still Engelsman [1984, 231] cites it simply as being § 30 in [Nic. Bernoulli 1720], apparently not noticing that while it is indeed § 30 in Johann Bernoulli’s Opera Omnia, it is numbered §29 in the original publication in the Actorum Eruditorum Supplementa, 7 (1721), pp. 310–312, because of a duplication of § 22). Now, a formula somewhat similar to (6.14) does occur in [Nic. Bernoulli 1720, 443] — namely, dq = Tq dy + R dy, for dx = pdy + q da, where dp = Tdx + Sdy + Rda and dq is the differential of q holding a constant; in modern notation, and noticing that holding a constant makes \( dq = \frac{{\partial q}} {{\partial x}}dx + \frac{{\partial q}} {{\partial y}}dy = \frac{{\partial q}} {{\partial x}}p dy + \frac{{\partial q}} {{\partial y}}dy, \) , this amounts to \( \frac{{\partial q}} {{\partial x}}p dy + \frac{{\partial q}} {{\partial y}}dy = q\frac{{\partial p}} {{\partial x}}dy + \frac{{\partial p}} {{\partial a}}dy, \) , whence \( p\frac{{\partial q}} {{\partial x}} + \frac{{\partial q}} {{\partial y}} = q\frac{{\partial p}} {{\partial x}} + \frac{{\partial p}} {{\partial a}} \) , that is, the condition of integrability of dx = p dy+q da. Not only these later developments are not present, but also Bernoulli does not use the formula at all as a criterion for integrability; rather, he uses it to obtain q, given p (i.e., to solve what Engelsman [1984] has called the “completion problem”). What really appears in Bernoulli’s derivation of that formula for the first time is something else, although essential for (6.13): the equality of mixed second-order differentials — Lacroix noticed this in [Montucla & Lalande 1802, 344].Google Scholar
  21. 71.
    In spite of Fontaine’s (and to some extent Nicolaus (I) Bernoulli’s) priority, it was Clairaut who communicated (6.13) to Euler [Engelsman 1984, 198].Google Scholar
  22. 76.
    These cones are made up of straight lines satisfying (6.23), but unlike what Taton [1951, 298] says, they do not satisfy (6.23) themselves. The whole point is that these equations belong to families of curves, not to surfaces.Google Scholar
  23. 84.
    In [Lagrange Calcul, 112] he abandoned the distinction in terminology, calling both kinds of equations either “prime equations” or “first-order derivative equations”, etc.Google Scholar
  24. 85.
    Not thoroughly complete, as he published three memoirs in the volumes of the Paris Academy for 1767 and 1768 (thus after [Fontaine 1764 ]). We may also notice the contradiction between the inclusion of “a few things that had appeared earlier”, namely in the memoirs of the Paris Academy for 1734 and 1747, and the first title of [Fontaine 1764 ], which mentions the unpublished character of the works contained within.Google Scholar
  25. 86.
    Condorcet’s researches would later evolve into a theory of integration in finite terms [Gilain 1988 ], which remained mostly unpublished: his main work on this was a large treatise of integral calculus which was only partly printed (152 of what would be about 1000 pages [Gilain 1988, 127]; according to Lacroix [Traité, 2nd ed, I, xxii–xxiii] those printed pages circulated at the time; but he was only able to study the whole manuscript in 1824 [Gilain 1988, 110].Google Scholar
  26. 87.
    Later, he would deny this inclusion [Lagrange Calcul, 372–381]. See section 9.5.3 below.Google Scholar
  27. 91.
    But not always: Legendre [1787, 340] gives two counter-examples in which the integrals thus obtained, although including an arbitrary function, are not as general as the “complete” one (because the functions involved have fewer arguments than the one in the “complete integral”).Google Scholar
  28. 92.
    Bossut [1798, II, 373–386] reports Lagrange’s method for integrating (quasi-)linear firstorder partial differential equations, but not his method of quasi-linearization, which might have motivated some reference to integrals with arbitrary constants instead of arbitrary functions.Google Scholar
  29. 95.
    We may also notice that Fontaine’s conception is very clear in [Monge 1785b], a memoir on ordinary differential equations.Google Scholar
  30. 97.
    When he later pays more attention to elimination of functions, it is to eliminate arbitrary functions from equations in more than two variables (see section 6.2.2) — something not in [Euler Differentialis, I, ch. 9].Google Scholar
  31. 99.
    [Lagrange Calcul, 168–177] does, explaining their existence in a way similar to Lacroix’s, although more detailed and generalized. But the first edition of [Lagrange Calcul] was first printed in 1801 [Grattan-Guinness 1990, I, 196], three years after [Lacroix Traité, II].Google Scholar
  32. 100.
    There is one detail related to this in which Lacroix’s and Laplace’s terminologies are different: Lacroix speaks of “complete integrals”, while Laplace [1772a] spoke of “general integrals”.Google Scholar
  33. 104.
    As will be seen below, singular solutions (“singular primitive equations”) are introduced in [Lagrange Fonctions] in a way that associates them to failures in certain power series. But it should be remarked that the adjective “singular” seems to have been associated with failures in more general power-series expansions (non-analyticity, in modern terms) only in the second edition of [Lagrange Fonctions] (dated 1813), and only in the title of chapter 5 — not in its text.Google Scholar
  34. 105.
    This is similar to an example in [Euler Integralis, I, §544].Google Scholar
  35. 107.
    More correctly, as [Lacroix Traité, II, 281–282] puts it: the functions which when equaled to zero yield those integrals/solutions. zero yield those integrals/solutions.Google Scholar
  36. 108.
    [Trembley 179091] is not always very easy to follow: his uses of the expression “particular integrals” are particularly unhelpful (see section 6.1.1).Google Scholar
  37. 109.
    [Legendre 1790] was only published in 1797, but it was already printed in 1794, along with the other memoirs in the Paris Académie des Sciences’ volume for 1790 — the devaluation of banknotes had prevented its sale in the meanwhile. Given the facts that Lacroix uses this memoir both here and when dealing with particular solutions of partial differential equations (see below), in a volume published in 1798, and that Lacroix had been elected a correspondent of the Académie in 1789, it is very likely that he had access to the printed memoir while still unpublished.Google Scholar
  38. 110.
    [Houtain 1852, 1181] claims that Legendre’s proof (and consequently Lacroix’s) rests on a vicious circle. However, I believe that at least in the case of Lacroix the purpose of the proof is not to demonstrate that a singular solution contains less than n arbitrary constants (something which was taken for granted in the 18th century), but rather a simpler consequence: that the finite (or primitive) equation obtained from it (that is, its integral) contains less than n arbitrary constants.Google Scholar
  39. 111.
    [Lagrange 1779, 613–614], addressing singular integrals, introduces this operator using the symbol δ and then suddenly invokes the theory of variations (not the equality of mixed partial differentials) for δdV = dδV.Google Scholar
  40. 116.
    It is possible to eliminate ϕ and ψ separately using the total differentials dp and dq, but these differentials are of second order, and so are the resulting equations, which are the closest one can have to first integrals [Lacroix Traité, 548–549].Google Scholar
  41. 118.
    With a safeguard about the possibility that Condorcet’s unpublished treatise might address the subject? In 1798 Lacroix might know its beginning (he knew it in 1810), but he certainly did not know yet the whole manuscript (see footnote 86 above).Google Scholar
  42. 121.
    Legendre (who as already mentioned, called “complete integral” one with an arbitrary function) considered here instead the variation δ relative to the arbitrary function [Legendre 1790, 235–236].Google Scholar
  43. 125.
    In a sentence added in the errata, Lacroix [Traité, II, 730] explains that this is why in the writings of the early analysts who dealt with integral calculus “to construct a differential equation” is often the same as to integrate it or to separate its variables.Google Scholar
  44. 129.
    In the second edition Lacroix is more direct in dismissing any usefulness of these constructions for approximation, and in explaining that they serve to prove the “reality” of differential equations (see section 9.5.3); in the second edition he also seems less convinced of the practical usefulness of the analytical version of Euler’s “general method” for approximating differential equations (see section 9.4.2). The third and later editions of [Lacroix 1802a] also suggest non-approximative purposes (see section 8.8.2).Google Scholar

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