1550078481-Ordinary_Differential_Equations__Roberts_

328 Ordinary Differential Equations

SOLUTION

Letting u 1 = y, u 2 = y<^1 l , and u 3 = y<^2 l, we obtain the desired equivalent
linear first-order system initial value problem

U~ = U2

u; = U3

U3 I = ( ln ( x^2 - 4 )) U1 + 3e-X U2 - -.-2U3 + x^2

smx

u 1 (2 .5) = - 3, u2(2.5) = 0, u3(2.5) = 1.2.

The function a 1 (x) = ln(x^2 -4) is defined and continuous on (-oo, -2) and

(2,oo). The function a 2 (x) = 3e- x is defined and continuous on (-00, 00).

The function a 3 (x) = -2/(sinx) is defined and continuous for x =f. mr where

n is an integer. And the function b( x) = x^2 is defined and continuous on

(-00,00). Since (2,7r) is the largest interval containing c = 2.5 on which

the functions a 1 (x), a 2 (x), a 3 (x), and b(x) are simultaneously defined and
continuous, (2, 7r) is the largest interval on which a unique solution to the
IVP (22) and the linear system above exists.

Notice that a linear n-th order differential equation is equivalent to a linear
system of first-order differential equations; likewise, a nonlinear n-th order dif-
ferential equation is equivalent to a nonlinear system of first-order differential
equations. Higher order system initial value problems may also be rewrit-
ten as equivalent first-order system initial value problems as we illustrated at
the beginning of this chapter for the coupled spring-mass system (3). The
following example further demonstrates this technique.

EXAMPLE 5 Converting a Higher Order System Initial Value

Problem Into a First-Order System IVP

In chapter 6, we saw that the position (x, y) of an electron which was
initially at rest at the origin and subject to a magnetic field of intensity H
and an electric field of intensity E satisfied the second-order system initial
value problem

(23a)

(23b)

x" = -HRy' +ER

y" = HRx'

x(O) = 0, x' (0) = 0, y(O) = 0, y' (0) = 0.

Write the second-order system initial value problem (23) as an equivalent
first-order system initial value problem.