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 P 1 ,, P n  be n given functions in C( J ) .  We define an operator L as the transformation

        L: C ( n ) ( J )C( J )   by  fL( f )= f ( n ) + k=1 n P k f ( nk )   = k=0 n P k f ( nk )

where P 0 =1  and f ( 0 ) =f .  Introducing the derivation operators D k , we have

        L= D n + k=1 n P k D nk   = k=0 n P k D nk

where D 0 =1 .  A LDE of order n has the form

        L( y )=R          (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.  In fact, dim[ N( L ) ]=n , the order of L  [see §8.5].