1550078481-Ordinary_Differential_Equations__Roberts_

180 Ordinary Differential Equations

By Theorem 4.4 there are at least n linearly independent solutions to the
DE (19) on the interval I. The following representation theorem shows that
there are at most n linearly independent solutions of (19) on I and it provides
a representation for every solution of the DE (19) in terms of a ny set of n
linearly independent solutions. The following theorem does not say that there
is only one set of n linearly independent solutions- the set specified in the
proof of Theorem 4.4- but that the maximum number of members in any
solution set which is linearly independent on I is n.

REPRESENTATION THEOREM FOR N-TH ORDER

HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS

Let an(x), an-i(x), ... , a 1 (x), ao(x) be continuous on the interval I and

let an(x) =f=. 0 for all x E J. If y 1 (x ), Y2(x), ... , Yn(x) are linearly independent

solutions on I of the n-th order homogeneous linear differential equation

and if y(x) is any solution of (21) on I , then

for suitably chosen constants Ci.

Proof: Let y(x) be a ny solution of the DE (21) on the interval I and let x 0 be

any point in J. Consider the function z(x) = C1Y1(x)+c2y2(x)+· · +cnYn(x)

where the Ci are arbitrary constants. By the superposition theorem z(x) is

a solution of the DE (21) on I. If we can choose the Ci so that

(22) z (xo) = y(xo), zCll(xo) = yCll(xo), ... , z(n-l)(xo) = y<n-l)(x 0 ),

then z(x) = y(x) for all x E J , since by the existence and uniqueness theorem

there is only one solution of the DE (21) which satisfies the initial condi-

tions (22). Differentiating the function z(x) successively n - 1 times and

then evaluating z(x) and its n - l derivatives at x 0 , we obtain the following

system of n equations in then unknowns ci:

Or, in matrix-vector notation,

y(xo)

yCll (xo)