# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

**Download**(direct link)

**:**

**370**> 371 372 373 374 375 376 .. 609 >> Next

9. (a) dv/dt = 9.8, v(0) = 0 (b) T = V300/4.9 = 7.82 sec

(c) v = 76.68 m/sec

10. (a) r = 0.02828 day—1 (b) Q(t) = 100e—0 02828t (c) T = 24.5 days

12. 1620ln(4/3)/ ln2 = 672.4 years

13. (a) Q(t) = CV(1 — e—t/RC) (b) Q(t) ^ CV = QL

(c) Q(t) = CVexp[— (t — t1)/RC]

14. (a) Q = 3(1 — 10—4Q), Q(0) = 0

(b) Q(t) = 104(1 — e—3t/10 ), t inhrs; after 1 year Q = 9277.77 g

(c) Q = — 3 Q/104, Q(0) = 9277.77

(d) Q(t) = 9277.77e—3t/10 , t inhrs; after 1 year Q = 670.07 g

(e) T = 2.60 years

15. (a) q' = —q/300, q(0) = 5000 g (b) q(t) = 5000e—t/300 (c) no

(d) T = 300 ln(25/6) = 7.136 hr

(e) r = 250ln(25/6) = 256.78 gal/min

Section 1.3, page 22

2, 6, 8, 14, 16, 19 1. Second order, linear 2. Second order, nonlinear

3. Fourth order, linear 4. First order, nonlinear

5. Second order, nonlinear 6. Third order, linear

15. r =-2 16. r =±1

17. r = 2, ---3 18. r = 0, 1, 2

19. r =-1, -2 20. r = 1, 4

22, 26 21. Second order, linear 22. Second order, nonlinear

23. Fourth order, linear 24. Second order, nonlinear

CHAPTER 2 Section 2.1, page 38

1. (c) y = ce—3t + (t/3) — (1 /9) + e—2t; y is asymptotic to t/3 — 1/9 as t ^ æ

1,2,3 2. (c) y = ce2' + t3e2t/3; y —> æ as t —> æ

3. (c) y = ce—t + 1 + t2e—t/2; y — 1 as t — æ

Answers to Problems

681

See SSM for detailed solutions to 4, 6, 7, 11, 13, 15, 18, 20.

21b, 24bc, 28

30, 32, 35, 36, 37

1,4

6, 10ac, 13, 15, 17a

17c, 19ac

4.

5.

6.

7

8 9

10

11

12

13

15

17

19

21

22.

23.

25.

27.

28. 29.

36.

37.

(c) 7 = (c/t) + (3 cos2t)/4t + (3 sin2t)/2; 7 is asymptotic to (3 sin2t)/2 as t ¦

(c) 7 = ce2t — 3et; 7 ^æ or -æ as t ^ æ

(c) 7 = (c - tcos t + sin t)/12; 7 ^ 0 as t ^ æ 2 -2 -2 (c) 7 = t2e 1 + ce 1 ; 7 ^ 0 as t ^ æ

(c) 7 = (arctan t + c)/(1 + t2)2; 7 ^ 0 as t ^ æ

(c) 7 = ce-1/2 + 3t - 6; 7is asymptotic to 3t - 6 as t ^ æ

(c) 7 =—te-t + ct; 7 ^æ, 0, or -æ as t ^ æ

(c) 7 = ce-t + sin2t - 2 cos2t; 7 is asymptotic to sin2t - 2 cos2t as t ^ æ

(c) 7 = ce-t/2 + 3t2 — 121 + 24; 7 is asymptotic to 3t2 - 12t + 24 as t ^ æ

7 = 3er + 2(t - 1)e2t

7 = (3t4 - 4¥3 + 6t2 + 1)/12t2

7 = (t + 2)e2t

7 =-(1 + t)e-‘/t\ t = 0

(b) 7 = —4 cos t + 5 sin t + (a + 4)et/2;

5 vv/o «. I 5

(c) 7 oscillates for a = a0

(b) 7 = -3et/3 + (a + 3)et/2;

(c) 7 ^ -æ for a = a0

(b) 7 = te-t + (ea — 1)e-t/1;

(c) 7 ^ 0 as t ^ 0 for a = a0

14.

16.

18.

20.

a0 =

= (t2 - 1)

e 2t/2 2

7

7 = (sin t)/t2

7 = t 2[(n2/4) - 1 - tcos t + sin t] 7 = (t - 1 + 2e-‘)/t, t = 0

a3

a0 = 1 /e

24. (b) 7 =-cos t/12 + n2a/4t2; a0 = 4/n2

(c) 7 ^ ³ as t ^ 0 for a = a0 (t, 7) = (1.364312, 0.820082) 26.

(a) 7 = 12 + 65 cos2t + gsin2t - 75 e-t/4;

65 + 65

(b) t = 10.065778 70 = -5/2

70 = -16/3; 7 ^-t^as t-

See Problem 2.

See Problem 4.

70 = -1.642876 7 oscillates about 12 as t

æ for 70 = —16/3

Section 2.2, page 45

1.

2.

3.

4.

5.

6.

7.

8. 9.

11.

13.

15.

17.

19.

3 ó2 - 2x3 = c; 7 = 0

3 T2 — 2ln|1 + x31 = c; x = — 1, 7 = 0

7-1 + cos x = c if 7 = 0; also 7 = 0; everywhere

3 7 + 72 - x3 + x = c; 7 = -3/2

2tan27 — 2x — sin2x = c if cos27 = 0; also 7 = ±(2n + 1)n/4 for any integer n; everywhere

7 = sin[ln |x| + c] if x = 0 and 17I < 1; also 7 =±1

j2 — x2 + 2(e7 — e-x) = c; 7 + e7 = 0

3 7 + 73 - x3 = c; everywhere

(a) 7 = 1/(x2 - x - 6)

(c) - 2 < x < 3

(a) 7 = [2(1 - x)ex - 1]1/2

(c) —1.68 < x < 0.77 approximately

(a) 7 = — [2ln(1 + x2) + 4]1/2

(c) -æ < x < æ

(a) 7 = — + 5\/4x2 - 15

(c) x > 2vT5 _________________

(a) 7 = 5/2 -Óx3 - ex + 13/4

(c) — 1.4445 < x < 4.6297 approximately

(a) 7 = [n — arcsin(3 cos2 x)]/3

(c) |x - n/2| < 0.6155

10.

12

1/2

(a) 7 = —-/2x — 2x2 + 4 (c) - 1 < x < 2 (a) r = 2/(1 - 2 ln0)

(c) 0 <9 < ë/e___________

14. (a) 7 = [3 - 2^1 + x2]"

(c) |x| < 2^5 16. (a) 7 = — (x2 + 1)/2 (c) -æ < x < æ (a) 7 = — + ³^65 - 8ex - 8e (c) |x| < 2.0794 approximately (a) 7 = [|(arcsinx)2]1/3

**370**> 371 372 373 374 375 376 .. 609 >> Next