Discussions of LDEs are more easily carried in the operator notations.
Let be the linear space of all real-valued functions continuous on an interval J, which can be unbounded. Let be the subspace consisting of all real functions whose first n derivatives are also continuous on J. Let be n given functions in . We define an operator L as the transformation
where and . Introducing the derivation operators , we have
where . A LDE of order n has the form
where R is some known function on J. A solution of the LDE is any function in that satisfies (8.6) on J.
Using the linearity of D, it is easy to show that every Dk, and hence L, are also linear. Hence, L defined above is called a differential operator of order n.
With each nonhomogenous equation is associated a homogeneous equation . Let h be the most general solution of the homogeneous equation and p be any particular solution of the nonhomogeneous equation. Using the linearity of L, it is easily shown that the most general solution of the nonhomogeneous equation is given by .
The set of all solutions of the homogeneous equation is the null space , also called the solution space of . The solution space is a subspace of . Although may be infinite, is always finite. In fact, , the order of L [see §8.5].