Linear differential equation

In mathematics, a linear differential equation is a differential equation of the form

Related Topics:
Mathematics - Differential equation

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:Ly = f,

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where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. 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. Therefore a fairly general form of such an equation would be

Related Topics:
Differential operator - Linear operator - Right hand side - Derivative

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: D^n y(x) + a_{n-1}(x)D^{n-1} y(x) + cdots + a_1(x) D y(x) + a_0(x) y(x) =f(x)

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where D is the differential operator d/dx (i.e. Dy = y' , D²y = y",... ), and the ai are given functions. Such an equation is said to have order n, the index of the highest derivative of f that is involved. (Assuming a possibly existing coefficient an of this derivative to be non zero, it is eliminated by dividing through it. In case it can become zero, different cases must be considered separately for the analysis of the equation.)

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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.

Related Topics:
Ordinary differential equation - Contracted - Tensor - Partial differential equation

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The case where f = 0 is called a homogeneous equation, and is particularly important to the solution of the general case (by a method traditionally called particular integral and complementary function). When the ai are numbers, the equation is said to have constant coefficients.

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