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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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1 å' + 1
å 2‘ + 1 å3
- f º1 - 1 å
\-1 å'- 1
\ 6å 3
å-----
2
2t
21 + º31
2t 1 3t
å 2‘ + 1 å3
- f åt + 1 å-2‘ 4 åt - 1 º-21 1 º1 - 1 º-21
1 º1 - º-21 + 1 å3‘\
å 21 - 2 º-1 - 1 º21/
12
12 å
-2å‘ + º-21 + º31
— 1 º1 + å
+ 2 º31
12. x =
/ -2 3/2
º-1 sin2t
Section 7.8, page 407
1. x = C^2) ^ + C2
3 x = Ñ1 Ê] å-' + C2
4. x = å 7 + C
2. x = Ì2) + c2
5. x = C1 ^ 4j å t + c2 ^ 1 ² å2t +
6. x = c1 I 1 j º21 + c2 I 0 j º-1 + Cf I 1 I å
1 j ²å21 + |o j º21
å
å
713
See SSM for detailed solutions to 9 11, 12, 14
15, 17a
17bcd
17ef
-33 -6 1
12. x = - 1 e-t/2 + - I 5 I e
-7t/2
13. x = cA J t + c2
14. x = c1 ( 1 ) f 3 + c2
Ï t-3ln t -( 0) t-3
16. (b)
^ e-'/2 +
4)^"2+(21e-'/2-
17. (b) x(1) (t) =
(c) x(2)(t) =
(d) x(3)(f) =
e2t
1
te2t + ^1j e2t
(t2/2)e2t + |l j te2t + |oj e2t
( 0 1 t + 2 \
(e) V(t) = e2t 1 t +1 (I2/2) + t
- 1 -t -(t2/2) + 3yl
/ 0 1 2 -3 3 2
(f) T = 1 1 0 , T-1 = 3 -2 -2
- 1 0 3 -1 1 1
J =
/2 1 0\
0 2 1
v0 0 2/
4
714
See SSM for detailed solutions to19abc
19d
1,2
3
4
18. (a) x(1)(t) = ^0j et (3)(t) - ‘
(d) x(3)(t) =
1
(e) W(t) = º1 0
4t
1
or et 0
2t
4t
(f) T =
J
^2 -3 -2t - 1
2 0\
4 0
2 2 2t 1
19. (a) J2 =
1
0
2 -2 -1
/1 0 0\
0 1 1
(0 0 1
X2 2X
0 X:
T-1
J3
(1 -1/2
0 1/4
2 -3/2 -1
0
0
X3 3X2
0 X3
J4
X4 4X
0 X4
(c) exp (Jt) = eXt
1t 0 1
(d) x = exp(Jt)x0
20. (c) exp(Jt) = eXt
(1 0 0\ 0 1 t
V0 0 1
21. (c)exp(Jt) = eXt
(1 t t2/2^
0 1 t
V0 0 1 /
Section 7.9, page 417
1 x=c^0et+c2(3)e-t+3 (0tet -1 (3)et+(0t - (1
2. x c
1
2 I -S) e 2t - ^Ë/Ý
2/3 '' + e-'
3. x = 1 (2t - §sin2f - ³cos2f + c1) ( 5cosf j 5V 2 2 1 \2 cos t + sin t)
+ ³(-1 - 1 sm 2t + 3 cos 2t + c2) ^- cos52 sin t) ( ) (1) 2t (0) -2t 1 (1) t
4 x = ÷(-4)e-3t + MUe2t- Øe~° + 2U1 e'
5. x = cM2) + c2
c1 12
-2
c2 1
-5t
7. x = c^j e3' + cJ-Ë e-' + 1 M) º'
2
715
See SSM for detailed solutions to 12, 14
CHAPTER 8
1ac, 5ac, 7ac
x = c I 1 ² å‘ + c2 ( 1 ) å f + (M å‘ + 2 I1 ) ²å‘
41/ ^ 2\ç/ \0j T \1
cuDå-|/2+c^_1
9. x = cJM å-|/2 + cJ_M å-21 + ¥ f j f -¥ 15 ) + ( 6 ³å1
2/ \ 4 / V2
4 W 3 \2 -V^ -1 -^2
11. x = (2 sin2 t + c1) ( 5cost \ + (- 1 t - 2 sin t cos t + c2) Ã 5 sin f ^
V2 1 ^2 cos t + sin t) v 2 2 2 \- cos t + 2 sin t)
12. x = r°ln(sin t) - ln(— cos t) - 21 + c,l ( 5cost j
L5 v 7 v 75 1J\2cosf + sin t)
+ [2ln(sin t) - 51 + c2] Ã 5 sin ^ ^
L5 v J 5 2 \- cos t + 2 sin t)
/ e-f/2COS11 e-f/2sin1 A tn ( sin11
13. (a) W(t) = /2 1 2 (b) x = å-(/× 2 ,
ó4å sin11 -4å f/2cos1ty \4 - 4cos21
14. x = c1 (1) f + c2 (f) f-1 - (2) + 1 (f) f - Q f ln f- 1 (4) 12
15.x=c1 (2)f2+c2 (2)t-1+(2)f+10 (1)f4 -1 (2
Section 8.1, page 427
1. (a) 1.1975, 1.38549, 1.56491, 1.73658
(b) 1.19631, 1.38335, 1.56200, 1.73308
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