1550078481-Ordinary_Differential_Equations__Roberts_

Answers to Selected Exercises

Chapter 7 SYSTEMS OF FIRST-ORDER

DIFFERENTIAL EQUATIONS

5. a. linear b. (0, oo)

555

c. (-oo, oo) Because y 1 , yz, y~, and Y2 are all defined and continuous

at x = 0, but the system of differential equations (25) is not defined

at x = 0. No, because the initial value problem (25) is undefined at

x = o.

7. a. nonlinear
   b. There is a solution on some subinterval of ( -7r /2, 7r /2) containing
   the point x = 0.


8. a. nonlinear
   b. There is a solution on some subinterval of (-3, 4) containing the
   point x = 0.
   c. x , -3+, x , 4-, Y1 , -2+, Y1 , +oo, Y2 , -oo, Y2 , 2-.


9. a. Let u1 = y, Uz = y(l), U3 = yC^2 ), and U4 = yC^3 ). Then

u~ = u2

u2 = U3

u3 = U4

u4 = -3xui + u~ - exu3u 4 + x^2 - 1

b. Let u1 = y and u2 = y'. Then

u~ = u2

I C k.

u 2 =--u2--smu1

m m

e. Let u 1 = y, u 2 = y', u3 = z, and u4 = z'. Then

u~ = u2

u2 = 2u1 - 3u4

u3 = U4

u4 = 3u2 - 2u3