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In mathematics, an ordinary differential equation or ODE is a differential equation containing a function or functions of one independent variable and its derivatives. The term "''ordinary''" is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions. ==Background== Ordinary differential equations (ODEs) arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, describing dynamical phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates),〔Mathematics for Chemists, D.M. Hirst, Macmillan Press, 1976, (No ISBN) SBN: 333-18172-7〕 biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes). Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler. A simple example is Newton's second law of motion — the relationship between the displacement ''x'' and the time ''t'' of the object under the force ''F'', which leads to the differential equation : for the motion of a particle of constant mass ''m''. In general, ''F'' depends on the position ''x''(''t'') of the particle at time ''t'', and so the unknown function ''x''(''t'') appears on both sides of the differential equation, as is indicated in the notation ''F''(''x''(''t'')). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ordinary differential equation」の詳細全文を読む スポンサード リンク
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