The problem of string vibration caused a debate among D'Alembert, Euler and daniel bernoulli about the nature of string function as initial condition.
This debate has at least two meanings: (1) let mathematicians realize the importance of non-analytic functions and reflect on the meaning of the word function.
(2) D. Bernoulli conjectures that the chord function can be expressed as the sum of infinite trigonometric series, thus opening the door to the so-called Fourier series. (2) n-body problem: Based on Newton's law of gravity, the interaction process of N planets is discussed, which is the N-body problem in celestial mechanics.
When n=2, Newton has completely solved and deduced Kepler's law of planetary motion, and there is no general solution to the problem. Therefore, a series of celestial problems have been studied, and Euler, Laplace and Lagrange have all made important contributions. By the end of19th century, through Poincare's new viewpoint, the qualitative study of differential equations began, which opened the field of so-called dynamic systems (chaos is one of them).
In addition, the so-called Laplace equation (that is, (6)) is derived by considering the total gravity of the planet, and the same idea also appears in electromagnetism.
The important and far-reaching sources of differential equations are mainly physics and geometry. Besides the equations listed above, there are Euler and Naville-Stokes equations in fluid mechanics, Einstein equation in Einstein's general theory of relativity, Schrodinger equation in quantum mechanics, Dirac equation, geometric geodesic equation, minimal surface (submanifold) equation and so on.
Quite a few differential equations can be derived from a systematic viewpoint, which is called the variational method of function space (plus principle of least action).
In addition, when solving PDE problem, we can use symmetry to separate variables and turn the problem into ODE problem.
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