8.5. The Existence-Uniqueness Theorem

Theorem 8.3:  Existence-Uniqueness Theorem.

Let P 1 ,, P n  be continuous functions on an open interval J and let L be the linear differential operator

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

where D ( 0 ) =1  and P 0 =1 .  Let x 0 J  and let k 0 , k 1 ,, k n1  be n given real numbers.  Then there exists 1 and only 1 function y=f( x )  that satisfies both the homogeneous equation L( y )=0  on J and the initial conditions

        f ( j ) ( x 0 )= k j   for j=0,1,,n1

Proof

Proof of this will be obtained as a corollary of a more general theorem for systems of differential equations to be discussed in Chapter 10.  A specialized version for LDEs with constant coefficients is given in §9.9.

Comment

The initial conditions can be represented by a vector in Rn with components

( f( x 0 ), f ( x 0 ),, f ( n1 ) ( x 0 ) )

This is also called the initial- value vector of f at x0.  The uniqueness theorem then implies only f( x )=0  can satisfy an initial-value vector that equals to O.