1550078481-Ordinary_Differential_Equations__Roberts_

326 Ordinary Differential Equations

interval about c provided in some generalized rectangle R in xu1 u2 ... Un-

space, the function fn(x, u 1 , u2, ... , Un) = g(x, u1, ... , Un) and the n partial

derivatives 8fn/8ui = 8g/8ui, i = 1, 2, ... , n are all continuous functions.

S u b st1tu · t• mg c 1or u 1 , u 2 ,. .. ,un · m t erms o f y , y (l) , ... , y (n-l) , we no t e th a t

the initial value problem (18) will have a unique solution on some small inter-

val centered about x = c provided the function g(x, y, yU), ... , y(n-l)) and the

partial derivatives 8g/8y, og/fJyUl, ... , og/oy<n-l) are all continuous func-

tions of x, y, yU), ... , y(n) on some generali zed rectangle R in xyyU) ... y<n-l)_

space. Thus, we have the following existence and uniqueness theorem for the
gener al n-th order initial value problem (18).

EXISTENCE AND UNIQUENESS THEOREM FOR THE

GENERAL N-TH ORDER INITIAL VALUE PROBLEMS

Let R be the generalized rectangle

{(x,y,y(ll, ... ,y<n-l)) I a< x < /3 and 'Yi< y(i-l) < 8i,i = 1,2, ... ,n}

where a, /3, "fi, and 8i are all finite real constants. If g(x, y , y(l), ... , y<n-l))

. is a con t. muous f unc t. ion o f x, y, y (l) , ... , y (n-l) m · R , i ·f ug !'.:\ /!'.:\ uy, ug !'.:\ /!'.:\ uy <^1 l , ... ,

og/oy<n-l) are all continuous functions of x, y, y(l), ... , y(n-l) in R, and if

(c, d 1 , d 2 , ... , dn) ER, then there exists a unique solution to the initia l value

problem

(18a) y(n) = g(x, y, y<l)' ... 'y<n-1))

(18b)

on some interval I = ( c - h, c + h) where I is a subinterval of (a , /3) and

the solution can be continued in a unique manner until the boundary of R

is reached.

Recall that then-th order differential equation y(n) = g(x, y , y(l), ... , y<n-l))
is linear if and only if g has the form g(x, y, y(l), ... , y(n-l)) = a 1 (x )y +
a 2 (x)y(l) + · · · + an(x)y<n-l) + b(x). So, the linear n-th order initia l value
problem

(20a) y(n) = a1(x)y + a2(x)y(l) + · · · + an(x)y(n-l) + b(x)

is equivalent to the linear system initial value problem