1550078481-Ordinary_Differential_Equations__Roberts_

Answers to Selected Exercises 541

then y(x) _, +oo as x _, +oo. If a solution is negative for x > 0,

then y(x) _, -oo as x _, + oo. If a solution is positive for x < 0, then

y(x) _, +oo as x _, -oo. If a solution is negative for x < 0, then

y(x) _, -oo as x _, - oo.

11. The functions y = 0 and y = 3 are solutions. For y > 3 solutions are

strictly increasing and asymptotic to y = 3 as x _, - oo. For 0 < y < 3

solutions are strictly decreasing, asymptotic to y = 3 as x _, -oo and

asymptotic to y = 0 as x _, +oo. For y < 0 solut ions are strictly

decreasing and asymptotic toy= 0 as x _, + oo.

13. The function y = 0 is a solution. All other solutions are increasing.
    There a re no relative minima or relative maxima. Solutions below y =
    0 are asymptotic to y = 0 as x , +oo. Solutions above y = 0 are
    asymptotic to y = 0 as x , -oo.


14. The direction field is undefined for y ::::; - x. Solutions increase for

y > - x + 1 and decrease for - x < y < - x + l. Relative minima

occur where y = - x + l.

17. The direction field is undefined on and outside the circle x^2 + y^2 = 15.

All solutions inside the circle are strictly increasing.

Exercises 2.2 Fundamental Theorems

l. (-3, d) and (5, d) for all d 3. (0, d) for all d and ( c, 0) for all c

5. (c,d) for all c and -2 < d < 1 7. (c,O) for all c 9. (c, 1) for all c

11. (c,d) where c :'.'.'. 0 and d < 4 and (c,d) where c::::; 0 and d > 4

13. ( -oo, 00)


14. (0, 2)
    15. (-00,00) 17. ( - oo, 00)


15. No, because fy is undefined at (0, 1).

19. (7r/2, 37r/2)

Exercises 2.3 Solution of Simple First-Order Differential Equations

Exercises 2.3.1 Solution of y' = g(x)

3 2 1

l. y=

2

x +x-

2

; (-00,00) 3. y = -2cosx -1; (-00,00)

5. y=l+lnlx-11; (1,oo)