is amathematicalequationfor an unknownfunctionof one or severalvariablesthat relates the values of the function itself and itsderivativesof variousorders. Differential equations play a prominent role inengineering, physics,economics and other disciplines.Differential equations arise in many areas of science and technology: whenever adeterministicrelationship involving some continuously varying quantities (modelled byfunctions) and their rates of change in space and/or time (expressed as derivatives) isknown or postulated. This is illustrated inclassical mechanics, where the motion of a body is described by its position and velocity as the time varies.Newton's Lawsallowone to relate the position, velocity, acceleration and various forces acting on the body andstate this relation as a differential equation for the unknown position of the body as afunction of time. In some cases, this differential equation (called anequation of motion)may be solved explicitly.An example of modelling a real world problem using differential equations isdetermination of the velocity of a ball falling through the air, considering only gravityand air resistance. The ball's acceleration towards the ground is the acceleration due togravity minus the deceleration due to air resistance. Gravity is constant but air resistancemay be modelled as proportional to the ball's velocity. This means the ball's acceleration,which is the derivative of its velocity, depends on the velocity. Finding the velocity as afunction of time requires solving a differential equation.Differential equations are mathematically studied from several different perspectives,mostly concerned with their solutions, the set of functions that satisfy the equation. Onlythe simplest differential equations admit solutions given by explicit formulas; however,some properties of solutions of a given differential equation may be determined withoutfinding their exact form. If a...
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