1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations

(19a )

U~-1 = Un

(19b)

EXAMPLE 3 Converting an n-th Order Initial Value Problem

to an Equivalent System Initial Value Problem

Write the n-th order initial value problem

y (4) = 7 x2y + y(l) y (3) _ ex(y(2))3

y(O) = 1, y(ll(o) = -1, y<^2 l(O) = -2, y<^3 l(o) = 4

as an equivalent system initial value problem.

SOLUTION

325

In this instance, g(x, y, y(l), y<^2 ), y<^3 l) = 7 x^2 y+y(l)y(^3 ) - ex(y<^2 l)^3. Letting
u 1 = y, u2 = y(l), u3 = y<^2 l , and u 4 = y(^3 ), we obtain the desired equivalent
system initial value problem

I

U1 = U2

I

U 2 = U3

U~ = U4

U4 I =^7 X^2 U1 + U2U4 - e x3 U3

Notice in syst em (17) that 8fif8yj = 0 for i = 1, 2, .. .,n - 1 and
j = 1, 2, .. .,n but j ~ i + l. And also 8fif8yi+ 1 =1fori=1, 2, .. .,n-l.

Thus, all of the n(n - 1) partial derivatives 8fif 8yj i = 1, 2, ... , n - 1 and

j = 1, 2, ... , n are defined and continuous functions of x, u 1 , u2, ... , Un in
all of xu 1 u 2 ... Un-space. Applying the fundamental existence and uniqueness
theorem to system (17), we see that it will have a unique solution on a small