Differential equation

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Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

A differential equation is a mathematical equation that relates a function to its derivatives. Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions may be determined without finding their exact form. Pure mathematics considers solutions of differential equations. The theory of dynamical systems emphasizes qualitative analysis of systems described by differential equations. If no self-contained formula for the solution is available, many computer-driven numerical methods approximate solutions within a given degree of accuracy.


  • Almost all of fluid dynamics follows from a differential equation called the Navier-Stokes equation. But this general equation has not, in practice, led to solutions of real problems of any complexity. In this sense, the curve of a baseball is not understood; the Navier-Stokes equation applied to a base ball has not been solved.
    • Robert Adair The Physics Of Baseball (1990) (2nd Edition - Revised, 1994) Ch. 2, The Flight Of The baseball, p. 22.
  • I ought also to mention [Jacobi]'s papers on Abelian transcendants; his investigations on the theory of numbers... his important memoirs on the theory of differential equations, both ordinary and partial; his development of the calculus of variations; and his contributions to the problem of three bodies, and other particular dynamical problems. Most of the results of the researches last named are included in his Vorlesungen über Dynamik.
  • The meaning of the differential equation now follows:
    expresses the claim that the rate of change in ... is proportional at to itself.
    And this makes sense. How fast a colony of bacteria will grow is contingent on the... number of bacteria on hand and the relative percentage of bacteria engaged in reproduction. ...
    Equations are... acts of specification in the dark; something answers to some condition. ...Specification in the dark corresponds to the...process by which a sentence in which a pronoun figures—He smokes—acquires the stamp of specificity when the antecedent... is dramatically or diffidently revealed—Winston Churchill, say, or a lapsed smoker seeking an errant cigarette in a bathroom.
    The differential equation describing uniform growth admits a simple but utterly general solution by means of the exponential function
    The number is an irrational number lying on the leeward side of the margin between 2 and 3 and playing, like , a strange and essentially inscrutable role throughout all of mathematics; exponentiation takes to a power... in this case... specified by A and t. The constant k has an interpretation as the problem's initial value... some... (weight or mass) of bacteria. ...
    as time scrolls backward or forward in the... imagination, provides a running account of growth or decay...
    This is in itself remarkable, the temporal control achieved by what are after all are just symbols, quite unlike anything else in language or its lore or law, but when successful, specification in the dark achieves an analysis of experience that goes beyond any specific prediction to embrace a universe of possibilities loitering discreetly behind the scenes.
    • David Berlinski, The Advent of the Algorithm: the Idea that Rules the World (2000)
  • [W]e find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances... This memoir concluded with an invitation to mathematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions.
  • [Newton] teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem.
  • Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english.
  • The problem of three bodies has been treated in various ways since the time of Lagrange, but no decided advance towards a more complete algebraic solution has been made, and the problem stands substantially where it was left by him. He had made a reduction in the differential equations to the seventh order. This was elegantly accomplished in a different way by Jacobi in 1843.
  • We should speak of a dialectics of the calculus... the problem element in so far as... distinguished from the properly mathematical element of solutions. Following Lautman... a problem has three aspects: its difference in kind from solutions; its transcendence in relation to the solutions... and its immanence in the solutions which cover it, the problem being the better resolved the more it is determined. Thus the ideal connections constitutive of the problematic ([Platonic] dialectical) Idea are incarnated in the real relations which are constituted by mathematical theories and carried over into problems in the form of solutions... like the discontinuities compatible with differential equations.
  • In general, a differential equation arises whenever you have a quantity subject to change. ...Strictly speaking, the changing quantity should be one that changes continuously. ...However, change in many real life situations consists of a large number of individual discrete changes, that are miniscule compared with the overall scale of the problem, and in such cases there is no harm in simple assuming that the whole changes continuously.
  • If the idea of physical reality had ceased to be purely atomic, it still remained for the time being purely mechanistic; people still tried to explain all events as the motion of inert masses; indeed no other way of looking at things seemed conceivable. Then came the great change, which will be associated for all time with the names of Faraday, Clerk Maxwell, and Hertz. The lion's share in this revolution fell to Clerk Maxwell. He showed that the whole of what was then known about light and electro-magnetic phenomena was expressed in his well known double system of differential equations, in which the electric and magnetic fields appear as the dependent variables. Maxwell did, indeed try to explain, or justify, these equations by intellectual constructions. But... the equations alone appeared as the essential thing and the strength of the fields as the ultimate entities, not to be reduced to anything else.
    • Albert Einstein, "Clerk Maxwell's Influence on the Evolution of the Idea of Physical Reality" in Essays in Science (1934)
  • Mr Gregory devoted to it a chapter of his work, and noticed particularly some of the more remarkable applications of definite integrals to the expression of the solutions of partial differential equations. It is not improbable that in another edition he would have developed this subject at somewhat greater length. He had long been an admirer of Fourier’s great work on heat, to which this part of mathematics owes so much; and once, while turning over its pages, remarked to the writer,—“ All these things seem to me to be a kind of mathematical paradise."
    • Robert Leslie Ellis The Mathematical and Other Writings of Robert Leslie Ellis. (1863) ed., W. Walton. p. 198-99
  • A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
    The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
    The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
    Properties of the variance equation are of great interest in the theory of adaptive systems.
    • Rudolf E. Kálmán, New results in linear filtering and prediction theory (1961) p. 95. Article summary as cited in: "Rudolf Emil Kalman", MacTutor History of Mathematics archive, 2010.
  • Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.
    • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 427.
  • The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived...
    A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.
    • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p. 442.
  • At the end of the year 1820 the fruit of all the ingenuity expended on elastic problems might be summed up as—an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars and plates, and the definition of Young's modulus. But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of a section of a beam, producing a resultant, was a step towards a theory of stress; the use of differential equations for the deflexion of a bent beam and the vibrations of bare and plates, was a foreshadowing of the employment of differential equations of displacement; the Newtonian conception of the constitution of bodies, combined with Hooke's Law, offered means for the formation of such equations; and the generalization of the principle of virtual work in the Mécanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics.
  • Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way. ...the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks.
  • The differential equation of the first order

    ...prescribes the slope at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction."
  • When [Born and Heisenberg and the Göttingen theoretical physicists] first discovered matrix mechanics they were having, of course, the same kind of trouble that everybody else had in trying to solve problems and to manipulate and to really do things with matrices. So they had gone to Hilbert for help and Hilbert said the only time he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert didn’t know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six month earlier if they had paid a little more attention to him.
    • Constance Reid, Hilbert (1996) p. 182. Also quoted in Max Jammer, The Conceptual Development of Quantum Mechanics (1966) pp. 207-208. Jammer cites the original reference: Edward Condon, 60 Years of Quantum Physics, Physics Today (1962) Vol. 15, 10, 37, pp. 37-49.
  • Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations.
  • [D]ifferential equations... represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both. ...[S]ince Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns. ...[T]theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram... Loosely translated... "It is useful to solve differential equations."
  • Simulators set up the required system of interdependences, usually between electrical potentials or voltages as variables, by means of valve-amplifiers and electrical networks. Since the voltage across a capacitance is proportional to the integral of a current, that across an inductance to the first derivative of a current, and that across a resistor to the current itself, it is possible to arrange a network of electrical elements, with amplifiers and feeds-back where necessary, so that a given linear differential equation is caused to relate an ’output’ voltage to an ’input’ voltage. Thus a given linear system of interdependences can be simulated, either directly or in any convenient transformation. If non-linear relationships are required there is no universally applicable simple device, but there do exist a great variety of non-linear elements with non-linear characteristics that are known and to some extent; adjustable. These include non-linear resistors... and the characteristic curves of thermionic valves, of rectifiers and discharge vessels and of magnetic materials. Limits may be set by the use of neon tubes that become conducting when a certain voltage is exceeded, or by relays, and so on
  • The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoulli's Programma.. if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =, is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever.

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