is an equation in which the derivatives of a function appear as variables. Many of the fundamental laws of physics, chemistry, biology and economics can be formulated as differential equations. They express the relationship involving the rates of change of continuously changing quantities modeled by functions and are used whenever a rate of change (the derivative) is known but the process originating is not. The solution of a differential equation is usually a function whose derivatives satisfy the equation. The question then becomes how to find the solutions of those equations.

The mathematical theory of differential equations has developed together with the sciences where the equations originate and where the results find application. Diverse scientific fields often give rise to identical problems in differential equations. In such cases, the mathematical theory can unify otherwise quite distinct scientific fields. A celebrated example is Fourier's theory of the conduction of heat in terms of sums of trigonometric functions, Fourier series, which finds application in the propagation of sound and electromagnetic fields, optics, elasticity, spectral analysis of radiation, and other scientific fields

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