1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 321

CONTINUATION THEOREM

FOR SYSTEM INITIAL VALUE PROBLEMS

Under the hypotheses of the fundamental existence and uniqueness theo-

rem, the solution of the initial value problem (11) can be extended until the

boundary of R is reached.

The generali zed rectangle R mentioned in the fundamental theorem can be
enlarged in all directions until one of then+ n^2 functions f i, i = 1, 2,... , n
or 8 Ji/ ayj, i, j = 1, 2, ... , n is not defined or not continuous on a bounding
"side" of R or until the bounding "side" of R approaches infinity. Thus,
the continuation theorem guarantees the existence of a unique solution to
the initial value problem (11) which extends from one bounding "side" of R
through the initial point ( c, dI, d 2 , ... , dn), which is in R , to another bounding
"side" of R. The two bounding "sides" may be the same "side."
According to the continuation theorem the solution to the IVP (12) through
(1, -2, 4) can be extended uniquely until it reaches a boundary of RI· Thus,
the solution can be extended until at least two of the following six things
occur: x ---; -A, x ---; B, YI ---; - C, YI ---; -E < 0, Y2 ---; -D, or
y 2 ---; E. Enlarging RI by letting A, B, C, D , and E approach +oo and letting
E ---; o+ (the hypotheses of the fundamental theorem and continuation theorem
regarding fi and 8 Ji/ ayj are still valid on the enlarged generalized rectangle),

we find the conditions become x ---; - oo, x ---; oo, YI ---; -oo, YI ---; o-,

Y2 ---; -oo, or Y2 ---; oo. The solution of the IVP (12) is

(13)

(Verify this fact.) The functions YI(x) and Y2(x) and their derivatives are
simultaneously defined and continuous on the intervals (-oo, 0) and (0, oo ).
Since c = 1 E (0, oo), the set {YI(x), y2(x )} is a solution to (12) on (0 , oo).
From the solution we see that the conditions which limit the interval of ex-

istence and uniqueness in this case are x ---; oo, YI ---; o- , and Y2 ---; oo- the

latter two occur simultaneously as x ---; o+.

Linear system initial value problems- system initial value problems in which

each function fi ( x, YI, Y2, ... , Yn), i = 1, 2, ... , n in the system is a linear func-

tion of YI, y 2 , ... , Yn- are an important special class of initial value problems.

We state a separate existence and uniqueness theorem for linear system ini-

tial value problems due to the special results that can be obtained for these

problems regarding the interval of existence and uniqueness of the solution.