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In mathematics, linear differential equations are differential equations having solutions which can be added together in particular linear combinations to form further solutions. They can be ordinary (ODEs) or partial (PDEs). The solutions to (homogeneous) linear differential equations form a vector space (unlike non-linear differential equations). ==Introduction== Linear differential equations are of the form : where the differential operator ''L'' is a linear operator, ''y'' is the unknown function (such as a function of time ''y''(''t'')), and the right hand side ''f'' is a given function of the same nature as ''y'' (called the source term). For a function dependent on time we may write the equation more expressly as : and, even more precisely by bracketing : The linear operator ''L'' may be considered to be of the form〔Gershenfeld 1999, p.9〕 : The linearity condition on ''L'' rules out operations such as taking the square of the derivative of ''y''; but permits, for example, taking the second derivative of ''y''. It is convenient to rewrite this equation in an operator form : where ''D'' is the differential operator ''d/dt'' (i.e. ''Dy = y' '', ''D''2''y = y",... ''), and the ''An'' are given functions. Such an equation is said to have order ''n'', the index of the highest derivative of ''y'' that is involved. A typical simple example is the linear differential equation used to model radioactive decay.〔Robinson 2004, p.5〕 Let ''N''(''t'') denote the number of radioactive atoms in some sample of material 〔Robinson 2004, p.7〕 at time ''t''. Then for some constant ''k'' > 0, the number of radioactive atoms which decay can be modelled by : If ''y'' is assumed to be a function of only one variable, one speaks about an ordinary differential equation, else the derivatives and their coefficients must be understood as (contracted) vectors, matrices or tensors of higher rank, and we have a (linear) partial differential equation. The case where ''f'' = 0 is called a homogeneous equation and its solutions are called complementary functions. It is particularly important to the solution of the general case, since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called ''particular integral and complementary function''). When the ''Ai'' are numbers, the equation is said to have ''constant coefficients''. 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「Linear differential equation」の詳細全文を読む スポンサード リンク
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