미분 방정식 ( differential equation , 微分方程式 )은 미지의 함수와 그 도함수간의 관계를 나타내는 방정식이다.
Anointing2011. 12. 7. 16:29
미분 방정식(微分方程式)은 미지의 함수와 그 도함수간의 관계를 나타내는 방정식이다.
미분 방정식의 차수는 방정식에 나오는 도함수가 몇 계 도함수까지 나오는지에 따라 결정된다.
미분 방정식은 유체역학, 천체역학등의 물리적 현상의 수학적 모델을 만들 때에 사용된다.
따라서 미분 방정식은 순수수학과 응용수학의 여러 분야에 걸쳐있는 넓은 학문이다.
미분 방정식의 목표는 다음 세가지 이다. 첫째, 특정한 상황을 표현하는 미분 방정식을 발견하는 것이다.
둘째, 그 미분 방정식의 정확한 해를 찾는 것이다.
셋째, 그 찾은 해를 해석하여 미래를 예측하는 것이다.
미분 방정식에 대해 해가 있어야만 하는지, 아니면 해가 유일한지 등의 문제도 중요한 관심사이다.
그러나 응용수학자, 물리학자, 엔지니어들은 대개 주어진 미분 방정식을 푸는 데에 관심을 두기 마련이고,
여기서 얻어진 해는 다리, 자동차, 비행기, 하수도 등을 만드는 데에 이용되고 있다.
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves 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. Only the simplest differential equations admit solutions given by explicit formulas; however, some 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 accuracy.
Lec 1 | MIT 18.03 Differential Equations, Spring 2006
