8.4. Linear Differential Equations Of Order N
Discussions of LDEs are more easily carried in the operator notations.
Let C  J  be the linear space of all real-valued functions continuous on an interval J,
which can be unbounded.
Let C  n   J  be the subspace consisting of all real functions
whose first n derivatives are also continuous on J.
Let P1 ,
, Pn be n given functions
in C  J  . We define an operator L as the transformation
L : C n  J   C  J 
by
0
where P0  1 and f    f .
f
n
n
k 1
k 0
L  f   f  n    Pk f  n k    Pk f  n k 
Introducing the derivation operators D k , we have
n
n
k 1
k 0
L  D n   Pk D n k   Pk D n k
where D  1 .
0
L y  R
A LDE of order n has the form
(8.6)
where R is some known function on J. A solution of the LDE is any function in
C  n   J  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 L  y   R is associated a homogeneous equation
L  y   0 . 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
h p.
The set of all solutions of the homogeneous equation is the null space N  L  , also called
the solution space of L  y   0 . The solution space is a subspace of C  n   J  .
Although dim C  n   J   may be infinite, dim  N  L   is always finite.
dim  N  L    n , the order of L
[see §8.5].
In fact,