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

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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13. a1 b2 --- a2b1 = 0
16. c cos t 17. c/x
18. c/(1 - x2) 20. 2/25
21. 3*/e = 4.946 22. p(t) = 0 for all t
26. If t0 is an inflection point, and ó = ô(³) is a solution, then from the differential equation
P(t0W(t0) + q (t0^(t0) = 0.
Section 3.4, page 158
1. e cos2 + ie sin2 = —1.1312 + 2.4717i
2. e2 cos 3 - ie2 sin3 = -7.3151 - 1.0427²
3. -1
4. e2cos(n/2) — ie2sin(n/2) = —e2i = —7.3891² 2cos(ln2) - 2i sin(ln2) = 1.5385 - 1.2779² n-1 cos(2 lnn) + i n-1 sin(2 lnn) = —0.20957 + 0.23959i
8. 7 = c1 t
10. 7 = c 12. 7 = c
14. 7 = c 16. 7 = c1
5.
6.
7.
9.
11.
13
15
17
18
19
20 21 22.
23.
24.
26.
35.
36.
37. 39. 41.
ó = c1et cos t + c1et sin t
7 = c1 7 = c1 7 = c1
e2t + c2 e-4t
e-3t cos2t + c2e-3t sin2t
-t cos(t/2) + c2e-t sin(t/2) ó = c1 e-t/2 cos t + c2e-t/2 sin t ó = 2 sin 21; steady oscillation ó = e-2t cos t + 2e-2t sin t; decaying oscillation ó = —et-7l/2 sin2t; growing oscillation ó = (1 + 2^3) cos t - (2 - V3) sint
e‘ cos V5 f + c2e^ sin \/5 t e-t cos t + c2e-t sin t cos(3t/2) + c2 sin(3t/2)
et/3 + c,e-4t/3
e 2t cos(3t/2) + c2e 2t sin(3t/2)
7 = /e 7 = V2e
t/2cos t + 2 e
t/2
sin t;
-(t-n/4)
sin t; decaying oscillation
(a) u = 2et/6 cos(V23 t/6) - (2/4l3)et/6 sin^V2! t/6)
(b) t = 10.7598
(a) u = 2e-t/5 cos(V34 t/5) + (7/V34) e-t/5 sin^V^4 t/5)
(b) T = 14.5115
25. (a) ó = 2e-t co^V^ t + [(a + 2)/V5] e-t si^V5 t (b) a = 1.50878
(c) t ={n — arctan [^\/5/(2 + a)]}/\/5 (d) n/\[b
(b) T = 1.8763
(a) ó = e at cos t + ae at sin t
2
x =
T = 4.3003;
e
t2/2 dt
(c) a = 1, T = 7.4284; a Yes, ó = c1 cos x + c2 sin x.
No
Yes, ó = c1 e-t2/4 cos(V3 12/4) + c2e-‘2/4 sin^V3 t2/4) ó = c1 cos (ln t) + c2 sin(ln t) ó = cj t-1 cos(2 ln t) + c2t-1 sin(2 ln t)
= 2, T= 1.5116
40.
42.
7 = c1^ 1 + c2t ,2
7 = c11 + c2t 1
Section 3.5, page 166
ó = c, el + c2 te^
,e-"2 + c2e3t/2
7 = c1 7 = c1 7 = c1 7 = c1
et cos 3t + c2et sin3t
t/4
+ c2e
e2t/5 + c2te2t/5 11. 7 = 2e2t/3 - 3'º2'/3,
12. ó = 2te
13.
7 as t
2.
4.
6.
8.
10.
00
t/3
-3t/2
+ c2te + c2te
t/3
-3t/2
7 = c1 e 7 = c1 e
ó = c1e3t + c2te3t ó = c1e-3t/4 ¦+ c2te-3t/4 ó = e-t/2 cos(t/:2) + c2e~
t/2
sin(t/2)
ó = —e-t/3 cos 3t + 9 e-t/3 sin3f.
—æ as t -æ
ó ^ 0 as t -
00
e
690
See SSM for detailed solutions to 14, 17ab
17cd, 19, 21, 25
27, 30, 31b 33, 35, 38
42
I,4, 6, 8
II, 13, 16 19a, 22a
¦2(f+1)
14. ó = 7e—2(f+1) + 5fe-
15. (a) ó = e—3f/2 — 2fe— 3f/2
(c) f0 = 16/15, Ó0 =
--5 e— 8/5
= - 0.33649
—3f/2.
b= --
u 2
y0
= 5e—4/5 = 2.24664
(d) ó = e— 3f/2 + (b + §)fe-
16. ó = 2et/2 + (b— 1)fef/2; b = 1
17. (a) ó = e—f/2 + 5 fe-t/2 (b) 0 = f,
(c) ó = e-t/2 + (b + ³ )fe—t/2
(d) fM = 4b/(1 + 2b) ^ 2 as b ^æ; yM = (1 + 2b) exp[—2b/(1 + 2b)] as b
OO
18. (a) ó = ae— 2t/3 + (3a — 1)fe—2f/3 23. y2(t) = f3 25. y2(f) = f— 1 lnf 27. y2(y) = cos x2 29. yJx) = x1/4e—^
32. ó = c1 e—^/2?X eSs2/2 ds + c2e—^/2
33. y2(f) = y() [ Ó— 2(s) exp — f p(r) dr
•"0 L Js0
34. y2(f) = f— 1lnf
36. y2 ( x) = x 39. (b) Ó0 + (â/Ü)Ó0 42. ó = c1f—1/2 + c2f—1/2 ln f
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