"[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."
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common in mathematical models and scientific laws; therefore, differe
"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."
"Maxwells equations had abstract mathematical qualities which were profoundly new and important. Maxwells theory was formulated in terms of a new style of mathematical concept, a extending throughout space and time and obeying coupled partial differential equations of peculiar symmetry. ...If they had taken Maxwells equations to heart as Euler took Newtons, they would have discovered, among other things, Einsteins theory of special relativity, the theory of s and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis. A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwells equations naturally lead."
"In the interval between the discovery of and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileos problem, and the related theories of the vibrations of bars and plates, and the stability of columns."
"Science is a differential equation. Religion is a ."
"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 ; 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."
"When [Born and Heisenberg and the Göttingen theoretical physicists] first discovered 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."