1550078481-Ordinary_Differential_Equations__Roberts_

(21a)

Systems of First-Order Differential Equations

u' 1

U~-l = Un

327

Applying the fundamental existence and uniqueness theorem for linear system

initial value problems to system (21), we see that there exists a unique solution

to (21)- equivalently to (20)- on any interval which contains the point c and

on which the functions a1(x),a2(x), ... ,an(x) and b(x) are simultaneously

defined and continuous. Hence, we have the following theorem.

EXISTENCE AND UNIQUENESS THEOREM FOR LINEAR

N-TH ORDER INITIAL VALUE PROBLEMS

If the functions a 1 ( x), a2 ( x),. .. , an ( x) and b( x) are all defined and con-

tinuous on some interval I which contains the point c, then there exists a

unique solution on the entire interval I to the linear n-th order initial value

problem

EXAMPLE 4 Determining the Interval of Existence and

Uniqueness of a Linear Initial V~lue Problem

Write the third-order, linear initial value problem

2y(2)

(22a) y(^3 ) = (ln(x^2 - 4))y + 3e-xy(l) - -.- + x^2

smx

(22b) y(2.5) = -3, y<^1 l (2.5) = 0, y(^2 ) (2.5) = 1.2

as an equivalent first-order system initial value problem and determine the
largest interval on which there exists a unique solution.