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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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Birkhoff, G., and Rota, G.-C., Ordinary Differential Equations (4th ed.) (New York: Wiley, 1989). Sagan, H., Boundary and Eigenvalue Problems in Mathematical Physics (New York: Wiley, 1961;
New York: Dover, 1989).
Weinberger, H., A First Course in Partial Differential Equations (New York: Wiley, 1965; New York: Dover, 1995).
Yosida, K., Lectures on Differential and Integral Equations (New York: Wiley-Interscience, 1960).
The following books are convenient sources of numerical and graphical data about Bessel and Legendre functions:
Abramowitz, M., and Stegun, I. A. (eds.), Handbook of Mathematical Functions (New York: Dover, 1965); originally published by the National Bureau of Standards, Washington, DC, 1964.
Jahnke, E., and Emde, F., Tables of Functions with Formulae and Curves (Leipzig: Teubner, 1938; New York: Dover, 1945).
The following books also contain much information about Sturm-Liouville problems:
Cole, R. H., Theory of Ordinary Differential Equations (New York: Irvington, 1968).
Hochstadt, H., Differential Equations: A Modern Approach (New York: Holt, 1964; New York: Dover, 1975).
Miller, R. K., and Michel, A. N., Ordinary Differential Equations (New York: Academic Press, 1982). Tricomi, F. G., Differential Equations (New York: Hafner, 1961).
Answers to Problems
CHAPTER 1 Section 1.1, page 8
1. y ^ 3/2 as t ^? 2. y diverges from 3/2 as t ^ ?
3. y diverges from -3/2 as t ^? 4. y 1/2 as t ^?
5. y diverges from —1/2 as t ^? 6. y diverges from —2 as t ^ ?
7. y = 3 — y 8. y = 2 — 3y
See SSM for 9. y = y — 2 10. y = 3y — 1
detailed solutions to 11. y = 0 and y = 4 are equilibrium solutions; y ^ 4 if initial value is positive; y diverges
2, 4, 7, 11, 13, from 0 if initial value is negative.
15ab. 12. y = 0 and y = 5 are equilibrium solutions; y diverges from 5 if initial value is greater than 5; y ^ 0 if initial value is less than 5.
13. y = 0 is equilibrium solution; y ^ 0 if initial value is negative; y diverges from 0 if initial value is positive.
14. y = 0 and y = 2 are equilibrium solutions; y diverges from 0 if initial value is negative; y ^ 2 if initial value is between 0 and 2; y diverges from 2 if initial value is greater than
15. 2. (a) dq/dt = 300(10—2 — q 10—6) (b) q ^ 104 g; no
16, 21, 22 16. dV/dt = —kV2/3 for some k > 0.
17. (a) dq/dt = 500 — 0.4q (b) q ^ 1250 mg
18. (a) mv' = mg — kv2 (b) v ^ ymg/ k (c) k = 2/49
19. y is asymptotic to t — 3as t ^? 20. y ^ 0as t ^?
21. y ^ ?, 0, or —? depending on the initial value of y
22. y ^ ? or —? depending on the initial value of y
23. y ^ ? or —? or y oscillates depending on the initial value of y
679
680
Answers to Problems
See SSM for 24. y — — » or is asymptotic to V2t — 1 depending on the initial value of y
detailed solution 25. y — 0 and then fails to exist after some tf > 0
for 24 26. y —— or —» depending on the initial value of y
Section 1.2, page 14
1b 1. (a) y = 5 + (yo — 5)e—t (b) y = (5/2) + y — (5/2)]e—2t
(c) y = 5 + (yo — 5)e—2t
Equilibrium solution is y = 5 in (a) and (c), y = 5/2 in (b); solution approaches equilibrium faster in (b) and (c) than in (a).
2. (a) y = 5 + (yo — 5)et (b) y = (5/2) + [yo — (5/2)]e2t
(c) y = 5 + (yo — 5)e2t
Equilibrium solution is y = 5 in (a) and (c), y = 5/2 in (b); solution diverges from equilibrium faster in (b) and (c) than in (a).
3. (a) y = ce—at + (b/a)
(c) (i) Equilibrium is lower and is approached more rapidly. (ii) Equilibrium is higher.
(iii) Equilibrium remains the same and is approached more rapidly.
4ab, 6b, 4. (a) y(t) = ceat (b) y = ceat + (b/a)
8ab,10a, 13abc 5. y = Re^a‘ + (b/a)
6. (a) T = 2ln18 = 5.78 months (b) T = 2ln[9oo/(9oo — p0)] months
(c) po = 9oo(1 — e—6) = 897.8
7. (a) r = (ln2)/3o day—1 (b) r = (ln2)/N day—1
8. (a) T = 5 ln5o = 19.56 sec (b) 718.34 m
9. (a) dv/dt = 9.8, v(o) = o (b) T = V300/4.9 = 7.82 sec
(c) v = 76.68 m/sec
1o. (a) r = o.o2828 day—1 (b) Q(t) = 100e—0'02828t (c) T = 24.5 days
12. 162oln(4/3)/ ln2 = 672.4 years
13. (a) Q(t) = CV(1 — e—RC) (b) Q(t) — CV = QL
(c) Q(t) = CVexp[— (t — t1)/RC]
14. (a) Q = 3(1 — 1o—4Q), Q(o) = o
(b) Q(t) = 1o4(1 — e—3t^1o ), t inhrs; after 1 year Q = 9277.77 g
(c) Q = -3 Q/1o4, Q(o) = 9277.77
(d) Q(t) = 9277.77e—3(/1° , t inhrs; after 1 year Q = 67o.o7 g
(e) T = 2.6o years
15. (a) q = —q/3oo, q(o) = 5ooo 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 = o, 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 —» 1,2,3 2. (c) y = ce2t + t3e2t/3; y —» as t —»
3. (c) y = ce—t + 1 + t2e—t/2; y — 1 as t —»
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