Differential equations arise in many areas of science and technology; whenever a deterministic relationship involving some continuously changing quantities modeled by functions) and their rates of change (expressed as derivatives)is known or postulated. This is illustrated by classical mechanics, where the motion of a body is described by its position, velocity, acceleration and various forces ting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In many cases, this differential equation may be solved, yielding the law of motion. Canonical forms are mathematically studied from several different perspectives, mostly concerned with their solutions, functions that make the equation hold true. Only the simplest differential equations admit solutions given by explicit formulas. Many properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of curacy. Differential equations play a prominent role in engineering, physics, economics and other disciplines. The study of differential equations is a wide field in both pure and applied mathematics.
The subject of differential equations constitutes a large and very important branch of modern Mathematics. The study of differential equations is important because they x frequently used in application of Mathematics to problems in Science, e.g. velocity or acceleration in the study of motion generally supplies us with a differential equations, satisfied by unknown function. Differential equations play a prominent role engineering, physics,...
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