Editing Linear differential equation (section)

==Linear differential operator==
{{Main|Differential operator}}
A ''basic differential operator'' of order {{mvar|i}} is a mapping that maps any [[differentiable function]] to its [[higher derivative|{{mvar|i}}th derivative]], or, in the case of several variables, to one of its [[partial derivative]]s of order {{mvar|i}}. It is commonly denoted 
<math display="block">\frac{d^i}{dx^i}</math>
in the case of [[univariate]] functions, and 
<math display="block">\frac{\partial^{i_1+\cdots +i_n}}{\partial x_1^{i_1}\cdots \partial x_n^{i_n}}</math>
in the case of functions of {{mvar|n}} variables. The basic differential operators include the derivative of order 0, which is the identity mapping.

A '''linear differential operator''' (abbreviated, in this article, as ''linear operator'' or, simply, ''operator'') is a [[linear combination]] of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form<ref>Gershenfeld 1999, p.9</ref>
<math display="block">a_0(x)+a_1(x)\frac{d}{dx} + \cdots +a_n(x)\frac{d^n}{dx^n},</math>
where {{math|''a''<sub>0</sub>(''x''), ..., ''a''<sub>''n''</sub>(''x'')}} are differentiable functions, and the nonnegative integer {{mvar|n}} is the ''order'' of the operator (if {{math|''a''<sub>''n''</sub>(''x'')}} is not the [[zero function]]).

Let {{mvar|L}} be a linear differential operator. The application of {{mvar|L}} to a function {{mvar|f}} is usually denoted {{math|''Lf''}} or {{math|''Lf''(''X'')}}, if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a [[linear operator]], since it maps sums to sums and the product by a [[scalar (mathematics)|scalar]] to the product by the same scalar.

As the sum of two linear operators is a linear operator, as well as the product (on the left) of a linear operator by a differentiable function, the linear differential operators form a [[vector space]] over the [[real number]]s or the [[complex number]]s (depending on the nature of the functions that are considered). They form also a [[free module]] over the [[ring (mathematics)|ring]] of differentiable functions.

The language of operators allows a compact writing for differentiable equations: if
<math display="block">L = a_0(x)+a_1(x)\frac{d}{dx} + \cdots +a_n(x)\frac{d^n}{dx^n},</math>
is a linear differential operator, then the equation 
<math display="block">a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^{(n)}=b(x)</math>
may be rewritten
<math display="block">Ly=b(x).</math>

There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in {{mvar|y}} and the right-hand and of the equation, such as {{math|1=''Ly''(''x'') = ''b''(''x'')}} or {{math|1=''Ly'' = ''b''}}.

The ''kernel'' of a linear differential operator is its [[kernel (linear algebra)|kernel]] as a linear mapping, that is the [[vector space]] of the solutions of the (homogeneous) differential equation {{math|1=''Ly'' = 0}}.

In the case of an ordinary differential operator of order {{mvar|n}}, [[Carathéodory's existence theorem]] implies that, under very mild conditions, the kernel of {{mvar|L}} is a vector space of dimension {{mvar|n}}, and that the solutions of the equation {{math|1=''Ly''(''x'') = ''b''(''x'')}} have the form
<math display="block">S_0(x) + c_1S_1(x) + \cdots + c_n S_n(x),</math>
where {{math|''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub>}} are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval {{mvar|I}}, if the functions {{math|''b'', ''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}} are continuous in {{mvar|I}}, and there is a positive real number {{mvar|k}} such that {{math|1={{abs|''a''<sub>''n''</sub>(''x'')}} > ''k''}} for every {{mvar|x}} in {{mvar|I}}.