Respostas Livro Boyce-Diprima 9th Ed.

Transcrição

Respostas dos
Problemas
CAPiTULO 1
Seca() LI
2. y se afasta de 3/2 quando t oc
1. y 3/2 quando t
4. y -0 -1/2 quando r -0 oo
3. y se afasta de -3/2 quando t
x
6. y se afasta de -2 quando t co
5. y se afasta de -1/2 quando r - cc
8. y' = 2 - 3y
7. y . 3 - y
10. y' = 3y - 1
9. y' = y - 2
I I. y = 0 e y =4 sac) solucCies de equilibrio; v
4 se o valor inicial 6 positivo; y se afasta de 0 se o valor
inicial é negativo.
y = 0 c y = 5 sRo solucOes de equilibrio; y se afasta de 5 se o valor inicial 6 maior do que 5; y
0 se o
valor inicial é menor do que 5.
0 se o valor inicial c negativo; y se afasta de 0 se o valor inicial
v = 0 6 solucäo de equilibrio; y
positivo.
y = 0 c y = 2 siio solucaes de equilibrio:y se afasta de 0 se o valor inicial d negativo; y -0 2 se o valor
inicial esta entre 0 e 2;y se afasta de 2 se o valor inicial d maior do que 2.
18. (h)
(j)
17. (g)
19. (h)
20. (e)
16. (c)
(a) dq/dr = 300(10 -2 - q10'):q cm g. r em h
(h) q -> 104 g; nao
dl/ /tit = -kV 2/3 para algum k 0.
clu/dt = -0.05(u - 70); u sen°F, r em minutos
(a) del/ = 500 - 0,4q; q em mg, t cm h (b) q -0 1250 mg
,/mg/k
(a) nu,' = mg - kv 2(b) v
(c) k = 2/49
27. y 0 quando t -> 00
y d assintOtico a t- 3 quando t -0 co
y
oc, 0 ou -cc, dependendo do valor inicial de y
y -> cc ou -co, dependendo do valor inicial de y
y
co ou -oo ou y oscila, dependendo do valor inicial de y
y -> -oc ou d assintOtico a -,./2t - 1, dependendo do valor inicial de y
0 e então dcixa de existir depois de algum instante ti > 0
y
oo ou -00, dependendo do valor inicial de y
y
Secäo 1.2
(b) y = (5/2) + [yo - (5/2)]e-2'
I. (a) y = 5 + (yo - 5)e'
(c) y = 5 + (yo - 5)e-2'
A solucdo de equilibrio d y =5 em (a) e (c), y = 5/2 cm (b); a solucao tende ao equilibrio mais depressa
em (b) e (c) do que em (a).
2. (a) y = 5 + (yo - 5)ei
- (5/2)Je2'
(b) y = (5/2) +
(c) y = 5 + (yo - 5)e21
A solucäo de equilibrio é y = 5 em (a) e (c), y = 5/2 em (b); a soluctio se afasta do equilibrio mais depressa em (b) e (c) do que em (a).
555
556
RESPOSTAS DOS PROBLEMS
(a) y = ce-°` + (b/a)
(c) (i) 0 equilibrio e mais baixo e 6 aproximado mais rapidamente. (ii) 0 equilibrio 6 mais alto. (iii)
equilibrio permanece o mesmo e é aproximado mais rapidamente.
(b) Y' = aY
(a) ye =
(b) y = cen + (b/a)
(a) yi(t)=
y = cc"' + (b/a)
(b) T = 2 ln[900/(900 - po)] meses
(a) T = 21n 18 -14 5.78 meses
897,8
(c) po = 900(1 - e -6 )
(b) r = (ln 2)/Ndia-1
(a) r = (In 2)/30 dias-1
(b) 718.34 m
(a) T = 51n 50 -= 19.56 s
(b) T = .1300/4,9 7.82 s
(a) duldt = 9,8, v(0) = 0
(c) v *:-L, 76,68 m/s
(e) x = 245 In cosh(t/5) m
11. (b) v = 49 tanh(t/5) m/s
(f) T Z.= 9,48 s
(b) Q(t) = 100e-"2828'
12. (a) r 2.-4 0,02828 dia -1
(c) T 24,5 (has
1620 In(4/3)/ In 2 672,4 anos
(b) kr = In 2
(a) u = T + ( 1 0 - T)e-kr
6,69 h
(b) Q(t)-). CV =
(a) Q(t) = CV (1 - e-oRc)
ti)/RC]
exp1-(t
(c) Q(t) = CV
18. (a) Q' = 3(1 - 10- 4 Q), Q(0) = 0
Q(t) = 10 4 (1 - e-300I ),t ern h: depois de 1 ano Q 9277.77 g
Q' = -3Q/104 . Q(0) = 9277,77
(d) Q(t) = 9277,77e - "4 , t cm depois de 1 ano Q 670,07 g
(e) T 1-=., 2,60 anos
(b) q (t) = 5000e-o3"'
19. (a) q' = -q/300, q(0) = 5000 g
(d) T = 300 In(25/6) 428,13 min 7.136 h
(c) nao
(e) r = 250 In(25/6) 356,78 gal/min
Seciio 1.3
I. Segunda ordem, linear
3. Quarta ordem, linear
5. Segunda ordem, nao linear
15. r = -2
17. r = 2, -3
19. r -1, -2
21. Segunda ordem, linear
23. Quarta ordem, linear
CAPiTULO 2
2. Segunda ordem, nil() linear
4. Primeira ordem, nao linear
6. Terceira ordem. linear
16. r = ±1
18. r = 0,1,2
20. r = 1.4
22. Segunda ordem, nil° linear
24. Segunda ordem, nao linear
Seciio 2.1
(c) y = ce-3( + (03) - (1/9) + e -2`; y d assintOtica a //3 - 1/9 quando t -* 00
= e2t t3e2'/3;
oo
se quando t
(c) yc
cc
1 quando t
(c) y = ce' + 1 +1 2 e-72; y
(c) y (c/t) + (3 cos 20/4/ + (3 sen 20/2; y 6 assintOtica a (3 sen 20/2 quando t
oo
(c) y = ce2' - 3e'; y -> co ou -co quando t
0 quando t -> cc
(c) y (c - t cost + sen t)/( 2 : v
+ ce-12 ; y
co
0 quando t
(c) y
oo
0 quando t
) ; y
(c) y = (arctan t + c)/(1 4. -2,2
(c) y = ce-1 / 2 + 3t - 6; y e assintOtica a 3t - 6 quando t -* co
co
oo, 0, ou -oo quando t
(c) y = -te" +ct; y
oo
(c) y = ce' + sen 2t - 2 cos 2t; y e assintOtica a sen 2t - 2 cos 2t quando t
co
(c) y = ce- `/2 + 3t2 - 12t + 24; y 6 assintOtica a 312 - 12t + 24 quando t
14. y (t2 - 1)e-272
13. y = 3e` + 2(/ - 1)e2'
+
1)/121
y = (sen 01(2
2
1
6.
413
+
612
y
=
(3t4
15.
4,1
_ - t cos t +sent]
18. y = 1.--2 [(7 2 / 4+)
17. y = (t + 2)e2'
20. y = (1- 1 + 2e -`)It, t A 0
19. y = -(1 + 0e -7e, t A 0
y
00
P ESFOSTAS DOS P ROBLEMS 557
cos t + s sen t + (a + Deì2 ; no =
(b) y =
(c) y oscila para a = ao
(b) y = -3eti3 + (a + 3)e/2 ; ao = -3
-oo para a = ao
(c) y
(b) y = [2 + a(37 + 4)e 2"3 - 2e-'/2)/(37 + 4); ao = -2/(37 + 4)
0 para a = no
(c) y
(b) y = to - ` + ( ea -1)e - ' It; no =11e
(c) y -> 0 quando t -> 0 para a= ao
(b) y = -(cost)/t'- + 7 2 a/4t2 ; ao = 4/72
0 para a = a()
(c) y -> quando t
(b) y= (e' - e + a sen 1)/sen t; no = (e - 1)/sen 1
1 para a = ao
(c) y
28. yo = -1.642876
27. (1, y) = (1,364312;0,820082)
788 e -fict ; y oscila em torno de 12 quando t
(b) y = 12 + A cos2t + sen 2t - 7,3(c) r = 10,065778
yo = -5/2
31. yo = -16/3; y -> -oo quando t -> oo para yo = -16/3
40. Veja o Problema 4.
41. Veja a Problema 6.
oo
Seclio 2.2
2. 3y2 - 21n11 +x 3 1 = c; x A -1.y 5.1-- 0
1. 3y2 - 2x 3 = c; y¢ 0
+ cos x = c sey 0 0; tambërn y = 0; em toda parte
3y. + y2 - X3 ± x = c; y 0 -3/2
2 tan 2y - 2x - sen 2x = c se cos 2y r= 0; tamb6m y = ±(2n + 1)7r/4 para todo inteiro
em toda parte
y =sen[ln lx1 + cl se x A 0 e <1; tambêm y = ±1
8. 3y + y3 - x 3 = C; em toda parte
7. y2 - x 2 + 2(e - e -x ) = c; y + eY 0 0
(c) -2 < x < 3
(a) v = 1/(x2 - x - 6)
(c) -1 < x < 2
- 2x2 + 4
(a) y =
(c) -1,68 < x < 0,77 aproximadamente
10
x)e
t
11
(a) v = 12(1 (c) 0 < B <
(a) r = 2/(1 - 21n0)
(a) y = -[2 Im1 + x 2 ) + 41 L2(c) -00 < X < 00
(a) y = [3 - 2 31 +
112
+ .14x2 - 15
(a) y =
(a) y = - i(X2 + 1)/2 (a) y = 5/2 - 3X3 - ex + 13/4
- 8e - 8e- x
(a) y = - +
(a) y = Err - arcsen (3 cost x)1/3
+ 1]"
(a) y = [; (arcsen
(c) lx1 < 415.
(c) x >
(c) -cc <.x < 00
(c) -1,4445 < x < 4,6297 aproximadamente
(c) 1x1 < 2,0794 aproximadamente
(c) Ix - 7/21 < 0.6155
(c) -1 < < 1
y3 - 3y2 - x - x3 + 2 = 0, lx1 < 1
y3 - 4y - x3 = -1, 1X3 - 11 < 16/3 ou -1.28 < x < 1,60
y= -1/(x2 /2 + 2x - 1); x= -2
y = -3/2 + ,/2.r - e + 13/4; x = In 2
26. y tan(r2 + 2x); x = 25. y = -3/2 + jsen 2x + 1/4; x = 7r/4
SeY0 < 0
(a) y
seyo > 0; y = 0 seyo = 0; y-
(1)) T = 3,29527
(b) T = 2,84367
(a) y -+ 4 qu
quando t -> oo
(c) 3,6622 < yo < 4,4042
c
ad - bc
x= y+
,
In lay + + k; a 0 0, ay' + b 0
a
a31. (b) arctan (y/x) - In lx1 = c
(e) ly + 2x 1 3 IY - 2-r 1 = c
33. (b) ly - x1 = cly + 3x1 5 ; tamba.1 y = x
32. (b) X2 -1- y2 - cx3 = 0
(b) ly + -YI ly + ,1x1 2 =
(b ) 2x/(x + y) + In lx + yl = c; tambem y = -x
36. (b) x/(x + y) + In 1x1 = c; tatnbem y = -x 37. (b) 1x1 3 1x2 - 5y2 1 = c
38. (b) clx13 = 1y2 x2
558 RESPOSTAS DOS PROBLEMS
Sec5o 2.3
2. Q(t) = 120y[l — exp(—t/60)]; 120y
100In WO min Z-1' 460,5 min
50e-0.2 (1 — e -02 ) lb 7,42 lb
Q(t) = 200 + t — [100(200) 2 /(200 + 1) 2 ] lb, t < 300; c = 121/125 lb/gal;
lira c 1 lb/gal
_
t+4
5 ,20 2 sen t
(a) Q(t) = 6;.505io e -riso + 25 -5
(c) nivel = 25; amplitude = 25 2501/5002 0,24995
(c) 130.41 s
(a) (ln 2)/r anos (b) 9,90 anos (c) 8,66%
(c) 9,77%
(b)
$3930
(a) k(e" — 1)/r
(b) $102.965,21
10. (a) 589.034,79
k = $3086,64/ano; $1259,92
(b) $152.698.56
(a) t 135,36 meses
Qo exp(-0,000120970, t ern anos
(a) 0,00012097 ano- I(b)
(c) 13.305 anos
P = 201.977,31 — 1977,31e 0n2n , 0 < t < t1 6,6745 (semanas)
(b) r = 101n 2 6,9315
(a) r a.- 2,9632; nilo
(c) r = 6,3805
16. t = In V /In min L- 6.07 min
0,83
15. (b)
(a) n O ) = 2000/(1 + 0,048t)' 13(c) r 750.77 s
(a) WO= ce -k` + To + kT I (k cos cot + cosencot)/(k 2 + w2)
9,11°F; r ".:4' 3,51 h
R k /./k2 +w2 ; r = (1/co) arctan (w/k)
19. (a) c = k + (P/r) + [co — k — (P1r)]e-"Iv: lim c = k +(Pk)
T = (V In 2)/r: T = (V In 10)/r
Superior. T = 431 anos; Michigan. T = 71.4 anus; Erie. T = 6.05 anus; Ontario. T = 17,6 anos
( b) 5.129 s
21. (a) 45.783 m
(b) 5,248 s
20. (a) 50,408 m
(b) 5,194 s
(a) 48,562 m
(d) 256,6 s
(c) 15 ft/s
(b) 1074,5 ft
(a) 176.7 ft/s
(b) = (66/25) In 10 mi -1 :=4. 6.0788 mi
v
(a) duldx =
(c) r = 900/(11 In 10) s L-' 35,533 s
m
k vo
kvo )
m uo
nt 2g
— In 1 + —)
t„,
(a) x, -- In 1 + — +
mg
m g
k2
(b) r = — gt: sim
26. (a) v = —(mg/k) + Ivo + (mg/k)I exp( —kt/m)
(c) v = 0 para t > 0
27. (a) vL = 2a 2 g(p —
(b) e = 4ira3g(p — p')/3E
(c) k > 0,2394 kg/s
(a) 11,58 m/s
(b) 13,45 m
(a) v = Ri2g1(R + x)
(0) 50,6 0
30. (b) x = ut cos A, y = —gt 2 12+ut sen A + h
(d) —16L2 /(u2 cost A) + L tan A + 3 > H
(f) u = 106.89 ft/s, A = 0,7954 rad
e) 0,63 rad < A < 0,96 rad
(
31. (a) v (tt cos A)e -", w = —g/r + (u sen A + g/r)e-"
(b) x = it cos A(1—e-")/r, y = —gt/r+ (it sen A + glr)(1— e - ")/r + h
(c1) rt = 145,3 ft/s, A = 0,644 rad
32. (d) k = 2,193
1. t =
3. Q
2051
cos
Seca() 2.4
2. 0 < t < 4
1. 0 < t < 3
4. —oo < t < —2
3. 7r/2 < t < 37r/2
6. 1 < t <
5. —2 < t < 2
8. 1 2 + y2 <1
2t + 5y < 0
7. 2t + Sy >
9, 1 — ( 2 + y 2 > 0 Ou 1 — t 2 + y 2 < 0, r 0, y 0
0, v A 3
11. y
10. Em todo o piano
12. t tur para n 0, ±1, ±2 .... ; ye -1 13. y = ±iy,2) — 4t 2 se Yo A 0; It1 < IYoI/2
se yo #0; y = 0 se yo = 0;
y = [(1/yo) —
o interval° 6 Iti < 1/VT se yo > 0; —oo < t < oo se yo 0
0 011
I
y = Yo/12ty (+ lseyo
0; y = 0 se yo = 0;
o interval° 6 —1/2y ) < t < oo se yo 00; —oo < t < cc se yo = 0
RESPOSTAS DOS PROBLEMAS 559
y = ± 1,I i 111(1 + t 3 ) + yii; -[1 - exp(-3yZ/2)1" < t < co
y -). 3 se yo > 0: y = 0 se yo = 0; y -> -oo se yo < 0
19. y -4. 0 se yo 9; y -> co se yo > 9
18. y --> -oc se yo < 0; y -> 0 se yo > 0
y .- -oo se yo < y, ';',--' -0,019; caso contrario y 6 assintOtica a ,5-17
(b) Sim: faca t,,= 1/2 na Eq. (19) no texto
(a) Niio
(c) lyl < (4/3)' 2 =-- 1,5396
22. (a) y,(1) 6 uma soluctio para t . 2: yAt) 6 LIMa solucâo para todo t
(b) f n5o 6 continua cm (2, -1)
i
1
1
26. (a) y i (i) = -: Y2( t )
p(s)g(s)ds
= P0)
-- ,,,
AU) 29. y = r/(k + ere')
)J
1/2
5ct
5
±151/(2
+
28. y =
y = ±
EE / ( a
±
cee-2ff )JI/2
I 7
I
y = ± Ip 0 )
2
p(s)ds + c
. onde WO = exp(21 sent + 2 Tt)
to
y = 1(1 - e-2') para 0 < t < 1; y = -!;(e 2 - 1)e -2 ' para t >
y = C I para 0 < t < 1; y = e - ''''' para t > 1
1
Secii() 2.5
y = 0 ë. instavel
y = -alb 6 assintoticamente estiivel.y = 0 6 instavel
y= 1 e assintoticamente estavel, v = c v = 2 sao instaveis
5. y= 0 e assintoticamente estavel
y= 0 6 instavel
7. (c) y = [y„ (I - y„)kt]/[ I + (1 - y„)kt]
6. y = 0 6 assintoticamente estavel
y = 1 6 semiestavel
y -1 e assintoticamente estavel,y = 0 e semiestavel.y = 1 c instavel
y = -1 e y = 1 sac) assintoticamente estaveis,y = 0 6 instavel
y= 0 e assintoticamente estavel. y= b = la 2 6 instavel
y= 2 6 assintoticamente estavel, y= 0 6 scmiestavel.y = -2 6 instavel
y 0 e y= 1 silo semiestaveis
(a) r = (1/01n 4; 55.452 anos
(b) T = (1/ r)111[0(1 - a)/(1 - /3)a}: 175.78 anos
(a) y = 0 6 instavel,y = K e assintoticamcnte estavel
(b) Convexa para 0 < y < Kle, cOncava para Kle < < K
(b) y(2) -1' 0.7153K 57.6 x
kg
17. (a) y = K exp{ [In(yo/K)]e-"I
(c) r 2,215 anos
18. (b) (h/a),/ kla7r; sim
(c) k /a < 7ra2
19. (b k2/2g(aa)2
(d) Y„, = Kr/4 para E = r/2
(c) Y = Ey2 = KE[1 - (E/r)]
- (4h/ rK) j/2
(a) Y12 = K[1
(a) y= 0 6 instavel,y = 1 e assintoticamente estavel
( b) Y = Yof[Yo + (1 - yo)e't
(b) x = .vo exp[-ay0 (1 -
(c) xo exp(-ayo/f3)
(a) y = yoe- 13 '
131
(b) z = 1/[v + (1 - u)e 13 '1
(c) 0.0927
(a,b) a = 0: v = 0 6 semiestavel.
instavel.
a> 0: y= f e assintoticamente estavel e y=-
(a) a < 0:y = 0 6 assintoticamente estavel.
sit() assintoticamente estaveis.
a > 0: y = 0 é instavel; y = jti e y = (a) a< 0: y= 0 e assintoticamente estavel e y= a 6 instavel.
a = 0: y = 0 6 semiestavel.
a> 0: y= 0 C instavel ey=ad assintoticamente estAvel.
pq[e (q-l"' - 1]
28. (a) lim x(t) = min(p, q); x(t) - gea(q - P) '
(b) lira x(t) = p: x(t)
L
pat
pat +
-
p
560
RESPOSTAS DOS PROBLEMAS
Sectio 2.6
2. Niio é exata
1. x2 + 3x + — 2y = c
4. x2y2 + 2xy = c
3. x3 — x2 y + 2x + 2y 3 +3y = c
6. Nilo 6 exata
5. axe + 2bxy + cy2 = k
8. Não 6 exata
7. c' sett y 2y cosx = c; tambem y = 0
10.
y In x + 3x2 — 2y = c
c
—
3y
=
9. e'Y cos 2x + x 2
X 2 + y2 = c
12.
I 1 . Niio 6 exata
3/3
— 3x2 ]/2, Ix' < ,123y = [v
(24x3
+ X 2 - 8x — 16)1/2J/4, x > 0.9846
y [x —
16. b = 1: ezv + = c
15. 13 = 3; x2y2 + 2x3y = c
19. X 2 2 In lyl — Y -2 = c; tamb6m y = 0 20. e sen y + 2y cos x = c
22. x2ex sen y = c
21. xy2 — (y2 — 2y + 2)eY = c
25. i.t(x) = e3.'; (3x2y + y3 )e3i = C
24. (t) = exp f R(t) dt, onde r = xy
27. it (y) = y; xy + y cos y — sett y = c
; y = ce + 1 +
26. it (x)
tambem
y=0
—
In
IA
R(y) e2Y /y; xe2
c
=
30.
,u(y) = y2; xs 3xy +
y2
y
+
u(y) = seny; e sen
31. p(x,y) = xy; x3y + 3x2 + y3 =
y4 = c
Seclio 2.7
(b) 1.1975: 1,38549; 1.56491; 1,73658
(a) 1,2; 1,39; 1,571; 1,7439
(d) 1,19516; 1,38127; 1.55918; 1,72968
1.19631;1,38335;1.56200:
1,73308
(c)
(b) 1.105: 1,23205: 1,38578; 1,57179
1,364;
1,5368
1.1;
1,22;
(a)
(d) 1.1107; 1,24591; 1,41106; 1,61277
1,10775:
1,23873;
1,39793;
1,59144
(c)
(1)) 126; 1,5641; 1,92156: 2,34359
(a) 1,25; 1,54; 1,878; 2.2736
(c) 1,26551: 1.57746; 1,94586; 2,38287 (d) 1,2714; 1,59182: 1,97212; 2,42554
(a) 0.3; 0.538501; 0324821: 0,866458
0,284813; 0,513339; 0.693451; 0,831571
0.277920; 0,501813; 0.678949; 0.815302
( d) 0,271428: 0,490897; 0,665142; 0.799729
6. Converge para y 0: diverge para y < 0
5. Converge para y 0; Mio esta definida para y < 0
Converge
Converge para ly(0)1 < 2,37 (aproximadamente); diverge nos out ros casos
10. Diverge
9. Diverge
11. (a) 2,30800; 2,49006; 2,60023:2,66773; 2.70939; 2.73521
2,30167; 2,48263; 2.59352: 2.66227; 230519; 2,73209
2.29864; 2,47903; 259024; 2,65958; 2,70310: 2,73053
(c1) 2,29686; 2.47691: 2.58830; 2,65798; 2.70185; 2,72959
12. (a) 1,70308: 3,06605; 2,44030; 1.77204; 1,37348; 1.11925
1,79548; 3,06051; 2,43292; 1,77807; 1,37795; 1.12191
1.84579; 3,05769; 2,42905; 1,78074;1,38017; 1,12328
(d) 1,87734; 3,05607; 2,42672; 1,78224; 1,38150: 1,12411
13. (a) —1,48849; —0,412339; 1.04687; 1.43176: 1,54438; 1,51971
—1,46909; —0,287883;1.05351; 1.42003: 1,53000; 1.50549
—1,45865; —0,217545;1.05715; 1,41486; 1.52334;1,49879
(d) —1,45212; —0,173376; 1,05941; 1,41197; 1,51949; 1,49490
14. (a) 0,950517; 0,687550; 0,369188; 0,145990; 0.0421429; 0.00872877
0,938298; 0,672145; 0,362640: 0.147659; 0,0454100; 0.0104931
0,932253; 0.664778; 0,359567; 0.148416; 0.0469514; 0,0113722
(d) 0,928649; 0,660463; 0 357783; 0,148848; 0,0478492; 0,0118978
(a) —0.166134; —0,410872; —0,804660;4,15867
(b) —0,174652; —0,434238; —0,889140; —3,09810
Uma estimativa razotivel para y cm t = 0,8 6 entre 5,5 e 6. I\15o e possivel obter uma estimativa confiavel
em t = 1 clos dados especificados.
Uma estimativa razoavel para y em t = 2,5 6 entre 18 e 19. Nao 6 possivel obter uma estimativa
em t = 3 dos dados especificados.
19. (b) 0.67 < ao < 0,68
(b) 2,37 < ao < 2,38
11.
R E S BOSTAS DOS P ROBL EMS
Secáo 2.8
(u) + 2) 2 , w(0) = 0 2. dulds = 1 - (u) +3) 3 ,
2 k lk
C(/) = e 2' - 1
0,i(1) =(c)
1. dulds = (s + 1) 2
(a)
k=1
"
•
(-1)ktk
(a) 0„(t) -
E k! k =1
( a ) C(t) =
E (-1) +i
(c)
(k+1 /(k
k
1)!2"
w(0) = 0
_1
+ 2t - 4
(c) 11(11 n-c (Mt) =
k=1
tn+i
(a) .0„(t) = t
7. ( a )
(P„(t)
=
(c) lim„_,0,„(t) = t
(n + 1)!
E
k -1
t3
(a) ol (t) = 3;
1 .
t 3
=
t3
S + 7 .
03(i) = 3
14
02 0) = t -01 ( )
4
0 1 (0 = t.
04
= t
t2
2!
t-,
- 2!
02( t ) = t -
+
t3
+
3!
t 3 7t'- 3!
12. (a) 0 1 0) = -t - t- 0 2 (t)
= -I -
03(1) = -
t
-
1"
•
t
03(0 = t -
k=i
( 2
t4
(a) 0 1 (0 =
(a)
8. (a) c),(t) =
3 . 5 . • • (2k - 1)
O2(t)
E 2 . 5 8 ... (3k - 1)
13k-1
t2k
=
+
-
7 . 9 + 3 . 7 9 • 11 + (7 . 9) 2 • 15
31
4 + 4 • 7
-1
( 15
21 11
(7
3110
(
13
16. 10 4- 64 . 13
+ 0(t8),
7t 5
14/6
-I- 007).
+
5!
6!
311 6
- 00)
6!
t3
t2
1 3
t4t st°
- + - -F
- - - + 0(t7),
5 24
4
t 1t 4
3/ 54t6
- - - - 0(17),
2
12 20
45
2
6
t2t 4
7t5
04 (0 = -t - - + - + - + 0(17)
2
8
60 15
Seciio 2.9
y„ = (- (0,9)"yo; yn
= yo/(n + 1): y„ -+ 0
0 quandon -> co
co
quando
pc, quando n
y„ = yo,/(n + 2)(n + 1)/2; y„
se n= 4kou n= 4k-1:
Y°'
y„ se 11 = 4k - 2 ou n = 4k - 3;
-yo,
co
y„ nao tem limite quando n
12 quando n
y„ = (0,5)" (y0 - 12) + 12; y„
y,, = (-1)"(0.5)"(yo - 4) + 4; y„ -> 4 quando n
S. $2283.63
7. 7,25%
$258,14
(a) $804,62
(c) $1028,61
(b) $877,57
30 anos: $804,62/rnès; $289.663,20 total 20 anos: S899,73/mês;
$215.935,20 total
13. 9,73%
$103.624,62
16. (b) u„
-co quando n
co
(e) 3,5699
19. (a) 4,7263
(c) 3,5643
(b) 1,223%,
Probleinas Variados
1. y = (c/x 2 ) + (x3/5)
3. x 2 + xy - 3y - y3 = 0
5. x 2 y + xy2 + = c
7. x4 + x - y 2 - y 3 = c
9. x 2 y + x + y 2 = c
2. 2y+cosy -x- senx=c
4. y = -3 + cex-'
(1 - el-x)
6. y =
8. y = + cos 2 - cos x)/x2
+ x -I + y -21n = c; tambt3rn y = 0
10. x+
561
562
=c
13. y = tan(.r + x2 + c)
11. (x 3 /3) + xy +
15. y = c/cosh2(x/2)
17. y = ce3x - e2"
19. 2xy + xy 3 - = c
21. 2xy2 + 3x 2y - 4x + y 3 = c
e2t
23. v = - + C
3t
25. (x'-/y) + arctan(1y/x) = c
27. (x2 + y2 + OCT' = c
29. arctan(y/x) - In NA 2 + y2 = c
•
31. x3y2 + xy3 = -4
(a) y = t + (c - 0 - 1
(c) y =sent + (c cos t - sen t) - '
(a) v' - [x(t)+ b]v
+
12. y = ce' + e - `
14. x2 + 2xy + 2y2 = 34
16. e -x cosy + e2Y sen = c
x
,
18. y = e -2" f e' cis +
20. e
+ C Y =C
22. y3 + 3y - x 3 + 3x = 2
24. sen ysen2 x = c
26. CYR + In 1x1 c
28. x3 + x2y = c
30. (y2 /x3) + (y/x2 ) = c
1
32. - = -x f
ds + 2
Y S
(b) y = r' + 2t(c - (2)-I
(I) = exp[-(at 2 /2) - bt)
(b) u = [b f 12(t) dt + c]/ µ(t),
37. y = In t + c2 +1
+ c2 + In t
36. v =
> 0; v = (2/k) arctan(t/k) + c2 se
se
c,
=
c2
y = (1/k) In 1(k - t)/(k +1)1+
y =
+ c2 se c, = 0; tambem v c
- k2 <
um (ator integrante.
Stigestrio: /1(v) =
y = ±1., (t - 2c1)0 C/ c2 ; tamhem y = c
y
+ c 2 - to-'
c = c i f - In 11 + c l t I + c2 se 5,-L 0; y = ;t 2 + c, se c, = 0; tambem y = c
43. y = sen(/ + c2 ) = sen t + k, cost
42. y- = c, +
co1/2
y
=
c
45. t + c, =
=
2t;
tambem
(y - 2c1)(Y
44. 1.13 - 2cly + c2
47. e = ( t + c2)2 + c1
46. ylnlyl - y + c i y + t = c2 ; tambern y = c
49. y = 2(1 - 0-2
+ 1) 312 - 1
48. y =
y = 3 In t - In(t2 + I) - 5 arctan t + 2 + ; In 2 +
y=;t2 + 2.
CAPITULO 3
3.1
2. y = c l e - ' + c2e-2:
1. y = ci e + c2e -3'
4. y = c l e`12 + c2e'
3. v = c i et/2 + c2e-'13
6. y = cl e302 + c2e-31 i 2
5. y = c l + c2e-5'
y = c1 exp[(9 + 3.J)t/2] + c2 exp[(9 - 31-5-)t/2]
y = c1 expR 1 + 0)(1+ c2 expl(l - 173)t] 9. y = e': y -> oo quando t -> co
11. v = 12e0 - 8e02 ; y -> - cc quando t -> 00
10. y = ;-e-' - 1 e - 3; y -> 0 quando t -> cc
y = -1 - e - 3'; y -> - 1 quando t -> cc
y = A (13 + 5..ii) expR -5 + ../13)t/21+ A ( 13 - 5../i3) exp[( -5 - i13)t/2]; y -> 0 quando
Secao
DO
quando t -> co
- (2/./3) exp[(-1 - N/33)//4]; y
y -> co quando t -> oo
y = 10+20 + ;6,-(r+2)/2;
y -> -oo quando t -> cc
y" + y' - 6y = 0
18. 2y" + 5y' + 2y = ()
y = et + e-1 : o minim° y = 1 em t = 1n 2
y = - e' + 3e02 ; o maxim° e y = a ern t = 1n(9/4), y = Deny = 1n 9
21. a = -2
224 = - 1
y -> 0 para a < 0:y torna-se iliniitado para a > 1
y 0 para a < 1: nao existe a tal que todas as solucOes nao nulas se tornam ilimitadas
(a) y = 15- (I + 21:3)e -2t + 1(4 - 2/3)e(i2.
0,71548 quando t = s In 6 "=- 0,71670
(c) = 2
(b)
(a) y (6 + t3)e-2` - (4 + /3) e -"
t,,, = In[(12 + 3/0/(12 + 2/3)1, y„, =
+ 0)3 /(4 )4)2
L- 16,3923
p = 6(1 +
(d) t„, -> In(3/2), y,,, -> oo
(a)y = d/c (b) aY" +bY' + cY 0
(b)c < 0 (c) b < 0 c 0 <c < b2/4a
(a) b > 0 e 0 < c < b2 /4a
y =
y =
33) exp[(-1 +
563
Seca() 3.2
2. 1
1.
4. x2ex
3. e -4(
6. 0
5. -e21
8. -oc < t < 1
7. 0 < t < oc
10. 0 < t < oc
9. 0 < t < 4
12. 2 < x < 3:r/2
11. 0 < x < 3
15. A equacao é nao homogénea.
14. A equacao é nao linear.
17. 31e2: +ce2'
16. Nao
19. 51,1/(f ,g)
18. te` + ct
-4(t cost - sent)
v3 e y, formam urn conjunto fundamental de solucties se e somente se a,b 2 - a,b, 0.
-2 t
I t
+ je
y i (t) =
ie' y 2 (t) =
3"-1)-r' 1- e -11-1)
Y2(t) = - .214v1(t) = -1e -311-1) +
25. Sim
24. Sim
27. Sim
26. Sim
(b) Sim. (c) iy,(t).y.,(t)] e [y,(t),y,(t)] sac) con juntos fundamentais de solucOes; ry 2 (t),y 3 (1)] e b74(t),y,(t)]
nao sao
30. c cost
ct 2 e'
32. c/(1 - x2)
31. c/x
"=- 4.946
35. 3
34. 2/25
36. p(t) = 0 para todo t
40. Se t„ for urn porno de inllexao e se y = 5(t) for tuna solucao, entao. da equacao di ferencial, p(t,,)(p' (to ) +
(1(t())0((„) = 0.
Sim. y = cie-•212I c'2 dt + c2e 0/2
xu
Nao
*
1 cos x ,
1.1 (I) at, ,m cl it (X) = exp [- j (- + -) ad
v
x12(x) L Ao t
47.
.r2
2 - v2 ) 1. = 0
u"
+
3.r
Et'
+
(
1
+x
Sim. y = c l x -1 + c2x
49.
/1"
xu.
=
0
48. (I - x2 )p" - 2x/1' + a(a + 1)/./ = 0
51. As equacOes de Legendre e de Airy sao autoadjuntas.
Sim. y = I
'
c; f
Sec:10 3.3
2. e2 cos 3 - ie2 sen 3 -.4_ -7,3151 - 1,0427i
1. e cos 2 + ie sen 2 -1,1312 + 2,4717i
-1
e2 cos( g /2) - ie2 sen( g /2) -e2 i -7,38911
2 cos(In 2) - 2isen (In 2) 1,5385 - 1,27791
r l cos(2In>r)+ i;r -t sen (2 In ir) -0.20957 + 0.239591
8. y = c l ef cos
t
+ c2 esen
7. = clet cos t + c2 e`sent
10. y = c l e' cos t + c 2 e -i sen t
9. y = c l e21 + c2 C 4'
12. y = c 1 cos(3t/2) + c2sen(3t/2)
11. y = c l e -31 cos 2t + c2 e -3r sen 2t
14. y = c i e(13 + c2e-403
13. y =
cos(t/2) + c2 e - ' son (02)
cos(3t/2) + c2e-2rsen(3t/2)
15. y = c l e -112 cos t + c2 e'a sent
16. y =
17. y = sen 2t; oscilacáo regular
y = e 2r cost 2e -21 sen t; oscilacao decrescente
y = -e'-'T/2 sen 2t; oscilacao crescente
y = (1 + 20) cos t - (2 - 0) sen t; oscilacao regular
y = 3e-'12 cos t + ;e- ''22 sen t; oscilacao decrescente
y = ../e-(1-7`14) cos t +
e - (1-7/4 'sen t; oscilacao decrescente
(a) u = 2e1/6 cos(if t/6) - (2/03)e06 sen (
t/6)
(b) t = 10,7598
24. (a) u = 2e -1/5 cos(/34 t/5) + (7/ .01-1)e -"s sen (04' t/5)
(b) T = 14,5115
(b) a = 1,50878
25. (a) y =2e - ' cos ../5 t + [(a + 2)/A c' sen ./3 t
(c) t = (7r - arctanI20/(2 + a)))/,/3
(d)
(b) T = 1,8763
26. (a) y = e' cos t + ae-ai sen t
(c) a = a, T = 7,4284; a = 2, T = 4,3003; a = 2, T = 1,5116
fn
L._
564
35. y = c1 cos(In t) + c2sen(Int)
37. y = c1 t -1 cos(; In t) + ot -1 sen (; In t)
39. y = ci t2 + c,t3
36. y = c1 t-1 + c2t-2
38. y = ci t6 + c,t-1
40. y = cl t cos(2 In t) + c2 1 sen (2 In t)
42. y = ci t-3 cos(In t) + c2 t -3 sen (In t)
41. y = c i t + c2t-3
Sim, y = ci cos x + c2 sell x, x= f e-'212 dr
Niio
46. Sim. y = cl e-'214 cos(ij t 2 i4) + c2 e-'214 sen (i3 t2/4)
Seciio 3.4
+ c2 te-' 3
2. y =
4. y = ci e -3 " + c2 te-3"
6. y = cl e3' + c2 te3'
8. y = cl e -3r;4 c2te-3'''
7. y = c i e-0 + c2e-11
10. y = Cl/2 cos(t/2) + c2 e -ti2 sen (t/2)
9. y = cl e2'15 + c2te2t/5
y = 2e20 - 3le2r/3 , y -co quando t oo
y = 2te31 , y co quando t oo
y = -e -113 cos 3t + 9e-ti3 sen 3t, y -› 0 quando t --> 00
y = 7e -20' 1 ' + 5te-2('+'), y 0 quando t -› co
(b) t = -25
15. (a) y = e-3'I2 - te-3(/2
-1- -0,33649
to = 16/15, yo =
1. y = c l e + c2te
3. r = ci e - ` 12 + c2e3'I2
5. y c i e cos 3t + c2esen 3t
(b + 4)te -302 ; h = -;
y =
16. y = 2e02 + (b - 1)tel2-; b= I
17. (a) y = e-'12 + ite-'I2
y = P -4I2
(b) t it = 5, ym = 5e- 3.5
2.24664
(b + 1)te-t/2
1M! = 4b/(1 + 2b) -4 . 2 quando b -+ co; y m = (1 + 2b) exp[ -2b/(1 + 2b)]
quando b -> co
18. ( a ) Y = ae-21/3 -F (ia - 1)te-2'13
( b) a =
24. y 2 (t) = t•-2
23. y2 (t) = t3
26. y 2 (t) = re'
25. y2 (t) = t- I In r
28. y 2 (x) = x
27. y 2 (x) = cosx2
30. y 2 (x) = x -1/2 cos .v
29. y2 (x) = x' 'e-2ìi
Y
32. y = cl e-'x:' ,2 f ease/2 ds + c2 e -ix2/2
o
34. y 2 (t) = t- 1 In t
36. y 2 (x) = x
39. (b) Yo + (a/ b))/0
42. y = ci t-1/2 + c2 t -112 In t
44. y = ci t -1 + c2t-- 1 In t
46. V = c, t -2 cos(3 In t) + c2t -2 sen (3 In I)
00
33. y 2 (t) = y i (t) f yi-2 (s) exp [- f p(r) d rids
(0
35. y 2 (t) = cos t37. y 2 (x) = x -112 cos x
41. y = c,t2 + c2 t 2 In t
43. y = ci t + C4512
45. y = c, 1 3/2 C2t3/2 In r
Secao 3.5
y = c l e3( +c2 e - ' -
e2r
y = cl e-' cos 2t + c2 e -t sen 2t +sen 2t - 11 cos 2t
3. y = ci e3' + c2e-' +
+
4. y = c 1 + c,e-2' + t - sen 2t - cos 2t
y = ci cos 3t + c2 sen 3t + (9t 2 - 6t + 1)e3` +
y =
+ c2 te-' + t2e-'
y = ci e-1 + c2e-'/2 +12 - 6t + 14- sen t - cos t
y ci cos t + c2 sen t - It cos 2t - sen 2t
u = cl cos wot + c2 sen coot + (4 - (02 ) -1 cos Cot
u = ci cos wot + c2 sen coot + (1/2w0 )t sen coot
cos(il3 t/2) + c,e 'I2 sen (../T3 t/2) +b e' y =
13.
et - l e -2r t y = Cie-1 c.2e2t
+
L'
cos
2t
+
y
=
4te - 3e + r 3ei + 4
1
4
2
15.
1
14. y
,4
SC
16. y = +
-- re -'
17. y =- 2 cos 2t - 18- sen 2t - it cos 2t
sen 2t + te-' sen 2t
y = e-' cos 2t +
(a) Y (t) = t(A 0t4 + A IP A2I2 4131 -1- A 4) + t ( B0 t2 -1- Bit B2)e-3'
+ D sen 3t + E cos 3t
(b) Ao = 2/15, A 1 = -2/9, A, = 8/27, A3 = -8/27, 11 4 = 16/81. BO = -1/9.
B 1 = -1/9, B2 = -2/27, D -1/18, E = -1/18
y =
(a) Y (t) = Aot + A 1 + t(Bot + BO sen t + r(Dor + D i ) cos r
(b) Ao = 1, A l = 0, Bo = 0, B 1 = 1/4, Do = -1/4, D 1 = 0
(a) Y(t) = el (A cos 2t + B sen 2t) + (Dot + D I )e2 sen I + (Eot + El )e2' cos t
(b) A = - 1 /20. B = -3/20, Do = -3/2, D I = -5, Eo = 3/2, E1 = 1/2
(a) Y(t) = Ae -' + t(Bot2 + B 1 t + B2 )e-' cost + t(Dot2 + D I t + D2 )e-' sen t
(b) A = 3, Bo = -2/3, B 1 = 0, B2 = 1, Do = 0, Di = 1, D2 = 1
(a) Y(t) = Aot2 + A l t + A2 + t2 (Bot + B1)e2i + (Dot + D 1 ) sen 2t + (Eot + EI ) cos 2t
(b) Ao = 1/2, A l = 1, A2 = 3/4, Bo = 2/3. B 1 = 0, Do = 0, D I = -1/16.
E0 = 1/8, E1 = 1/16
(a) Y(t) = t (Aot 2 + At + A 2 ) sen 2t + t(B0t2 + B 1 t + B2 ) cos 2t
(b) Ao = 0, A 1 = 13/16, A2 = 714, Bo = -1/12, B I = 0, B2 = 13/32
(a) Y(t) = (Aor 2 + A l t + A 2 )e sen 2t + (8012 + B i t + B2 )et cos 2t +
e-' (D cos t + E sen t) + Fe
(b) Ao = 1/52, A l = 10/169, A2 = -1233/35.152, Bo = -5/52. 8: = 73/676,
B2 = -4105/35.152. D = -3/2, E = 3/2, F = 2/3
(a) Y(t) = t(Aot - AOC' cos 2t + t (Bot + B i )e-' sen 2t + (Dot + D; )e -2r cos t +
(Eot + E1 )e-2' sen r
(b) Ao = 0, A l = 3/16, Bo = 3/8, B 1 = 0, Do = -2/5, D 1 = -7/25. E0 = 1/5,
El = 1/25
(b) to =
+ (-le
28. y = c1 cos At + c, sen At + E t a „,/( A 2 — m272 )1 sen mar t
rn.I
t.
29" Y = 1 -(1 + 7/2) sen t - (7/2) cos t + (7/2)e',
30. y = I 1 - e-•-: sen 2t - 1e -' cos 2t,
-1(1 + 2 )e -' cos 2t - (1 + e `12)e-' sen 2t ,
Niio
34. y
I
0<t<7
t>7
0 < t < 7/2
t > 7/2
+ c2 e-` -
Seca° 3.6
1. Y(t) =
2. Y (r) =
3. Y (r) = ,1r2e-g
4. Y(t) = 2r2e12
y = C I cos t + c2 sen t - (cost)t) In(tan t + sect)
y = c i cos 3t + c2 sen3t + (sen 3t) In(tan 3t + sec 3t) —
7 . y = c
+
- e-2' In t
y = c 1 cos 2t c2 sen 2t + (sen2r) In sen 2t - it cos 2t
y = c i cos(t/2) + c: sen (t/2) + t sen (02) + 2[In cos(t/2) I cos(t/2)
10. y =
+ c2 te - et In(1 + t 2 ) + arctan r
11.
= cl e2' + c,e3' + f [e31-s) - e2(`-s)]g(s)ds
12. Y= c 1 cos 2t + c2 sen 2t + f I sen 2(t - s)1g(s) ds
14. Y(t) = -2/2
13. Y(t) = ; + f 2 In t
Y(t) = -2(2t - 1)e-1
15. Y(t) = 1(t - 1)e2t16.
17. Y(.v) = 1. x2 (In .0 318. Y(x) = - ix 112 cos x
20. Y(x) = X -1/2 f t -3/2 sen(x - t)g(t) di
19. Y(x) = f A:et -ffei' g(r) ell
( 1 - r)(
(b) y= yo cos t + y', sen t + sen (t - s)g(s)ds
to
i(
.
y = (6 - a) I f [e'''') - ea('-')]g(s) ds 25. y = it -1 14 e(`-'5) sen µ(t - s)g(s)ds
fii
to
i
29. y = ci t + c2t2 + 412 In t
26. y= f (t - s)eau '3 g(s) ds
f
f
y = c 1 t -I + C2r5 + i4i
32. y = c1 e` + c2 1 - ;(2r - 1)e-`
31. y = c1 (1 + t) + c2 e + 1 (t - 1)0
565
566 RESPOSTAS DOS PROBLEMAS
Seca() 3.7
It = 5 cos(2t - 6). S = arctan(4/3) 0,9273
u = 2 cos(t - 27r/3)
u = 2./3 cos(3t - S), S = -aretan(1/2) -0,4636
u siff cos(7rt - 6). S = n + arctan(3/2) 4,1244
= 8 rad/s, T = 7r/4 s, R = 1/4 ft
cos 8t ft, t ern s:
u
ems:
t = 7r/14 s
t
u ; sen 141 cm.
u = (1/44) sen (841) - cos(84 t) ft, t em s; w = 8f rad/s,
T = 7144 s, R = V11 288 L'" 0.1954 ft, S = 7r - arctan(3/Nif) 2.0113
Q = 10 -6 cos 20001 C, t em s
u = e -10'[2 cos(lig t) + (51.A) sen(4.A 01 cm, tem s;
tt = 4 f rad/s. T, = r 2"6- s, T,/T = 7/2.A -='" 1,4289, r :1=_, 0,4045 s
t) ft. terns; t = 7r. /2./31 s 1" :=-= 1.5927 s
u (1/8,/31)e-2: sen (2
= 3.87008 rad/s,
ttl' 0,057198e-m5 ' cos(3.87008 t - 0,50709) m, t ern s;
ttlak, = 3,87008/ ./175- 0.99925
Q = 10-6 (2e-5mr - 6.-1°°c') C; t em s
13. y = .
= 1.4907
r = ,M 2 + B 2 , r cos 6 = B. rsen 9 = -A; R = r; S = 0 +(4tt + 1)7/2,
= 0,1,2, ...
18. R = 103 C2
y = 8 lb•s/ft
22.
2n/ 31
20. vo < --ytt0/2m
24. k = 6, r = ±2,is
23. y = 5 lb .s/ft
1.73, min r 4,87
(d) 25. (a) r 41,715
(e) r = (2/y) In(400/, 4 - y2)
[110,4km - y 2 cos At + (2mvo + ytto)sen 11 I] /V 41:11? - y26. (a) 11(t) =
=4m(ku,i+ yuor, + Ittu,)1( 411n - y2)
(b)
plu" + pogu = 0, T = 27 3 pll pog
(e) horario
(a) u = f sen f t
(c)
27 t/8)
(a) = (16/J23)c" sen(117
- b v61117:
sen( 17fiTI )
(b) u = cos( N lzt)
k
(c) A 0,98, 7' = 6.07
32. (b) u = sen t, A 1. T = 27
(d) E = 0,2, A = 0.96. T = 5,90:
E = 0,3, A = 0,94. T = 5.74
(f) e = -0,1, A = 1.03. T = 6,55; E = - 0,2, A = 1.06, 7' = 6.90: E = - 0.3,
A 1,11,T =7,41
Seca() 3.8
1. -2 sen 8t sett t
2. 2 sen(t/2) cos(13t/2)
3. 2 cos(37rt/2) cos(nt/2)
4. 2 sen (7t/2) cos(t/2)
u" + 256u = 16 cos 3t. u(0) = b, 11'(0) = 0, it em ft. t em s
it" + lOu' + 98u = 2 sen (r/2), u(0) = 0, u'(0) = 0,03, u Cm in, t em s
cos 16t +
(a) u
cos 3t
(c) w = 16 rad/s
(a) u = 153!281[160e75' cos( 73 t) +
t) - 160 cos(t/2) +
e"-` sen
3128 sen(t/2)]
(b) Os dots primeiros termos sao transientes.
(d) co= 4./3- rad/s
- ( cos 7t
cos
8t)
=
'
2
rt = f="1
8
sen
sen
(t/2)
sen
(1502)
ft,
I erns
45
45
1/8, 7r/8, 7r/4, 37r/8 s
It= (cos 8t + sen 8t - 8t cos 80/4 ft, t em s;
(30 cos 2t + sen 2t) ft, t em s (a)
(b) m = 4 slugs'
u = (12/6) cos(3t - 37/4) nt, tent s
Fo(t - sett 0,
0<t<
Fo[(27r - t) - 3 sett tj,
n < < 27r
15. 11 =
-4F0 sen t,
27r < t < oo
' A palavra slug signitica Iesnza. mas, neste context°, ë 111113 unidade de massa no sistema ing1C's: é uma massa que acelera
1 pc por segundo ao quadrado quando sob a acao do uma forca de 1 libra: 1 slug= 14.5939 kg. (N.T.)
+ 3) C, t ems, Q(0,001) 1,5468 x 10-6;
10-6(e-4"' -
00
Q(0.01) 2,9998 x 10 -6 : Q(t) -* 3 x 10 -6 quando t
(a) u = [32(2 - (02 ) cos wt 80)sen wt]/(64 - 63w2 + 16w4)
(b) A = 8/,./64 - 63(02 + 16w4(d) w = 3./14/8 1,4031, A = 64/ ,,/127 I" 5.6791
(a) a = 3(cos t - cos wt )1 (w2 - 1)
(a) tt = [(co2 + 2) cos t - 3 cos cot]1(w2 - 1) + sen t
CAPITUL 0 4
Seciio 4.1
2 . t > 0 ou t < 0
1. -oo < t < co
4. t > 0
3. t > 1, ou < t < 1. ou t < 0
, -37/2 < x < -7/2. -7/2 < x < 1. 1 < x < 7/2, r/2 < x < 37/2....
-oo < x < -2. -2 < x < 2. 2 < x <
Linearmente independente
Linearmente dependente:f, (t) + 3f.(t)- 2f,(t) = 0
Linearmente dependente: 2f,(t) + 13f2(t) - 3f,(t) - 7 f,(t) = 0
11. 1
Linearmente independente
13. -6e-2'
12. 1
15. 6x
14. e -2(
17. sen2 t = 0
16. 6/x
1 (5) - z cos 2t
19. (a) ao[n(n - 1)(n - 2) • • 11+ a l Inut - 1) • • • 2]t + • • + a,,t"
(aor" +a1rr.- • • a,,)e'r
e'
sim.
e` e' e2'
) 0, -cc < t < co
,e2'
22. W(t) =
21. W(t) = ce -2'
24. W(t) = •/t
23. W(t) = c / t2
28. y = c I t- + c,t3 + c3 (1 +1)
27. y = ()el + c2 t + este'
Seciio 4.2
1 . f esi(:r14)+2.1
3. 3ei(T V2m.7)
5. 2eilIIIN/61+2rill
7. 1. 1 (-1 +
4. erli3T1214.2nrri
6. ,,./ed(sx/4,1 2/44:T/
8. 1 r4 -Mi/S ,
1,( -1 -
(./73 +
- (./.7; + i) I Ni:5:
9. 1,
-1. -i
y = c1 e' + c2ie' + c3t2e'
11. y =
+ c2re' +
14. y = c 1 + (7,1 + c3e2z + c,te2'
13. y = c l er + c,e2' +
+ e -15112 (cs cos !7 t + ct, sen t)
15. y = 1 cos t + sen t + e " = (c3 cos t + sen
y = c, e: +
+ c3 e2' + c4e2'
+ c6r2e-'
y = c1 ei + c2te' + c3t: e' + c4e -' +
y = ci + c2 t + c 3 e' + c4 e-: + c5 cos t + c8 sen t
y = c + c2e' + c3e2r - c, cos t + (75 sen t
20. y = c1 + c2 e2' + (c3 cos 0 t + c4 sen
t)
y = e'[(c t + c2 t) cos t + (c3 + c,t) sen II + c`[(cs + ca) cos t + (c7 + c8t)sen ti
y = (c 1 + c2 t) cos t + (c; + cot) sen t
23. y = cl ef + c2e(2+`f ± c3e(2-.3),
32)(
c3e(-2C2e(-2+,
24. y =
+
y =
/2 + c2 e- " 3 cos(t/0) + c3c-r:3 sen (t/O)
y = c 1 e3' + cze- 2: +
+ c4e(3-A`
y = c I C° + c2e-'14 c3 e -' cos 2t + c4e-' sen 2t
y = c l e -' cos t + c2e -: sen t + c3c2' cos(0 t) + c4 e-21 sen
t)
29. y = 2 -2 cos t + sen t
sen(t/./2) - es/12 sen (t1 N/2)
30. y = :15
31. y = 2t - 3
32. y = 2 cos t - sen t
- e--z/234. y =
33. y = le' --
+ P,e'12 cos t + e1/2 sen t
y 8 - 18e -'13 + 8e-r2
170- ,
sen t -
COS(
sen(f t)
y = 2i; cos t -
+
39
sen
t)
r
y = (cosh t - cos t) + (senh
(a) W(1) = c, uma constante (b) W(t) = -8 (c) W(t) = 4
39. (b) a t = c i cos t + c2 sen t+c3 cos .16 t+c4 sen f t
4
2
e
21/4eri/8
568
Sec5° 4.3
I. y e ' er + c 2 tel + c 3 e-' + Ite" + 3
y c l e` + c 2 e-' + c 3 cos t + c 4 sen t - 3t - sen t
+ 4(t - 1)
+ C2 COS t + C3 SCI1 t
y =
t
cos
3
e"
+
2
e'
+
c
y = ci + c
y = + c2 t + c3e-2 ( + c4e2i _ 3e'- t4
y = c, cos t + c2 sen t + c3 t cos t c4 t sen + 3 + ycos2t
y = + c 2 t + c 3 1 2 + c4 e-' + e'12 [c5 cos(0 t/2) + c6 sen(0 t/2)] + .; t4
y = + c 2 t + c 3 t2 + c4 e' + sen 2t + 16 cos 2t
cos 2t) +
y =
10. y = ( 1 - 4) cos t -(Zt+ 4) sen t + 3t + 4
I1. y = I + 1(t2 + 3t) - tei
49
+
+ e3' N cos 2t - 1.
y = - I cos t - sen t +
) sen 2t
Y(t) = t(A0t3 -F A l t2 + A2t + A 3 ) + Bt2e'
Y(t)= t(Aot + AOC' + B cos t + C sen t
Y(t) = At2 er + B cos t + Csen t
16. Y(t) = At'- + (But + B I )e' + t(C cos 2t + D sen 2t)
I 7 . Y (t) = t(Aot2 + A l t + A 2 ) + (Bot + B t ) cos + (Cot + CI ) sent
+ te' (C cos t + Dsen t)
Y(t) = Ae` + (Bot +
+ • • • + a„_l a + an
ko = a0, k„ = aoa" +
Seciio 4.4
y = + c2 cost + c3 sen t - In cos t - (sen t) In(sec t + tan 1)
y = c, + c 2 el + c3 e-' - 1; 1 2
3. y = c i e! + c,e" + c 3 e2' +
y = + c 2 cos t c3 sen t + In(sec t + tan t) - t cos t + (sent) In cos t
y = c l efc 2 cos t + c 3 sen t- 1, e-' cos
y= c 1 cos t + c2 sen t + c 3 t cos t + c4 t sen t - ll t2 sen t
y c i e' + c 2 cos t + sen t - (cos t) In cos t + (sen t) In cos t - cos t
- 2t sen t + -I e l
/ Coss I ds
2
8. y = +
c3e-r - In sen t + In(cos t + 1) +
f ( e' 1 sen s) els-
+ le" f (e'/ sen s) ds
c l = 0, c2 = 2, c3 = 1 em resposta ao Problema 4
c 1 = 2, c, = c3 =
c4 = em resposta ao Problema 6
c 1 = Z. c , = Z, C3 =
, to = 0 em resposta ao Problema 7
c, = 3, c2 = 0, c3 = -e ra , to = :112 em resposta ao Problema 8
Y(x) =...r4/15
Y(t)
[e" - sen (t - s) - cos(t - s)]g(s) ds
• (I)
Y(t) =
f senh (t - s) - sen (t - s)]g(s) ds
to
Y(t)
f e(-" (t - s) 2g(s)ds; Y(t) = -tet In 111
17. Y(x) = z f [(x / t2 ) - 2(x2 /t3 ) + (x3 It4 ) ],g(t) dr
xo
CAPITULO 5
Seca() 5.1
1. p= 1
3. p = oo
5. p= i
7. p = 3
2. p = 2
4. p= 2
6. p = 1
8. p = e
f_ltx2n+1
9.
E (2n
" + I)!
n.0
P = 00
10.
E;
,
n =o n !'
=
11. 1+
oc
12. 1 - 2(x + 1) + (x + 1) 2 , p = oo
(x - 1), p = 0o
13. Ec_i>"÷i 1)"
p=
14. E(-0"x", p = 1
1
n.0
n=1
ti
15. Ex", p = 1
16.
n=0
=
E(-1)"+ 1 (, _ 2)", p =1
n=0
02xn
1+2 2 x +3 2 x 2 +42x3±... _
y" = 22 + 32 •2x + 42 • 3x2 + 52 • 4x3 + • • + (n + 2) 2 (n 1)x" + • • •
y' = a l + 2a,x + 3a3x2 + 4a4x3 + • • • + (n + 1)a„.,.ix" + • • •
= Ena„x"- 1
n=
=
I
E(n +1)a,:xn
n.0
y" =2a, +6a3x 12a4x2 + 20a5x3 + • • • + (n +2)(n +1)an+,e + • • •
oc
n(n - 1)a„x" -2 = E(n -2)(n +1)a„_:an
n -2
n =0
ti
a„-- 2x"
22. E21.(n+)a,
2x"
n=2
n=0
24. E [(n + 2)(n +1)an+2 - n(n - 1)a„
23. E(n + 1)anx"
n=0
n=0
26. a l + E
25. E [(n + 2)(n + 1)(4,1.2 + ?lad?
n=0
27.
+1)an. 1. 1
+
a„ - 1 ]•r"
n=1
28. a„ = (-2)"aoln!, n = 1,2
[(n +1)fla„ 4. 1 + a„ Ix"
•
n=0
Seciio 5.2
I. (a) a„ +2 = a„/(n + 2)(n + 1)
x 2 x4
(b,d)
(x) = 1 +
+
xx:n
x6
+
•••=
x 3x•5 x 7
3! 51 71
+ 2)
2. (a) (4.4-2 =
=
) . 2 (x) = x + - + - + - +
E
n=U (211)1
4. 1)! = senh
E (2nY2"4-I
x2
.r6
= 1 + - +
2 2 4 + 2 . 4 • 6
x3
-3- +
+ 3 5. 7
(b.d) y
Yz(x) =
= cosh X
=
2n
x
z+'
n=o 2nn1
2nn!x?"÷
(2n +1)!
3. (a) (n + 2)a ,,f2 - an+1 - a„ = 0
(b) .Y1(0 =1 + .12. (x - 1)2 + 1(x - 1) 3 + •k(x - 1)4 + • • •
Y2(x) = - 1) + 1(x - 1) 2 + 1(x - 1) 3 + 1(x - 1)4 +
4. (a)
= -k 2 anl(n + 4)(n + 3): a: = a3 = 0
k6xi2
k 2x4k4.0
(b,d) y i (x) = 1
+
+
3 -4 3 . 4 . 7 . 8 3 . 4 . 7 . 8- 11 • 12
cc:
(_1)ni--1(k2x4)nt-I
34 7 8
1+
4
34)
nt=
k 2 X5
+4
k4x9
8 . 95 k6x13
4 • 5 • 8 • 9 • 12 • 13
(-1)'"-3(k2x4)'•'
L [1 + E
4
•
5
•
8
• 9 . • • (4n1 + 4)(4m + 5)
m=0
Sugestlio: alga n = 4n1 n a relacäo de recorrencia.M = 1, 2, 3, ...
5. (a) (n + 2)(n + 1 ) a ni-2 - n(n + 1)an.,. 1 + a„ = 0, n > 1; a2 = -la()
+ • . • , y2 (x) = x - i6 x3 --
(b) y i (x) = 1 - -1-x2 - .1x3 -
+•••
a0e-2'
569
(a) a„. f. 2 = —(n 2 — 2n + 4)an/i2( n + 1)(n +2)1. n 2; a2 = — (1 0. a 3 = —
— :46 x 6 + • • • .
(b) y i (x) 1 — .v 2 +
X7 + • • •
Y2( x ) = x — 4 Xi+ 160x5 —
n=
0.1.2....
1),
+
n
(a) an+2-= —a n/(
x4
.v 2
X6
(-1)nx2n
. =+
4
(b.d) y i (x) = I —
1. 3
1 .) •
1
=1 1 • 3 • 5 • • • (2n — 1)
(_1)nx2n+1
x.
X 3
x5
4 • 6 • • • (210
=
+ 2 . 44 - 2 • 4 6 +
Y2 (X) = X -
n=} 2
1 )(ri + 2),
(a) (4.4.2 = — [ ( n + 1) 2 an+1 + a„ + an.-1]1( 11
—(ao + a l )/2
(b) y i (x) I — 1(x — 1) 2 +( .Y - 1) 3 -
n = 1,2....
a2 =
- 1)4 + • • •
Y2 (X) = (X - 1) -(.Y- 1)2 +(.Y- 1) 3 - (X - 1) 4 + • • •
(a) (n + 2)(n + 1)a„+ 2 + (n — 2)(n —3)a„ = 0: n = 0, 1, 2, ...
(b) (x) = 1 — 3x 2 , Y2( x ) = x — x3/3
(a) 4(n + 2)an+2 — (n — 2)a„ = 0: n = 0.1.2....
( b , d ) ))1( x )
x2
1 —y 2(x) _
X3
1,
X5
x7
2240
4n (2n — 1)(2n + 1)
(a) 3(n +2)an+2 — (n + 1)a„ = 0: n = 0.1.2....
5 6
x4
x 2
3 . . (2n — I) ,
(b•d) yi(x) = I + +4 1 432 x 6 +
3" 2 . 4 • • • (2,1)
2
(2n)
2
16
8
2
x2"1
•• •
X7
X5 +
Y2 (X ) = X 1- -X3
3' • 3 . 5 • • (2,1+ I)
945
135
9
(a) (a + 2)(n + 1)an+2 — ( n + 1 )na„_ 1 + ( n —1)a„ = 0: n = 0, 1, 2....
X 2 .v 3.v 4x't
+
+
+ • • • +
+ • • •
y2(x) =
(11,d) yi(x) = 1 +
0,1.2,
...
=
0;
(a) 2(n + 2)(n + 1)a,.+2 + (a + 3)a„ =
3 • 5 • • • (2,z + 1) x2/1 4.
+ • • • + (-1)"
—
(b,d) (x ) = — 3 +
2"(2n)!
384
32
4
x 3x 5x 74 6 • • (211+ 2) ,
—+- ± • • • + (-I)" 2 n (2n + 1)!
Y2 (X) =
(a) 2(n • 2)(n + 1)an+2 + 3(tz +1)a„.. 1 + (a +3)a„ = 0; n = 0, 1, 2....
— 2) 3 + X - 2) 4 + • • •
— 2) 2 +
(b) yi (x) = 1 —
(.v — 2) 3 + 4-(x - 2) .= + • • •
y7(x) = (X. - 2) — IOC - 2) 2 +
=
6
<0,7
(c) cerca de
(a) y = 2 + x + X2 -I- X3 + X4 + • • -
(c) cerca de lx1 <0.7
- (1, x 4 + • • •
(a) y = —1 + 3x + x 2 —
(c) cerca de lx1 < 0,5
(a) y 4 — x — 4x 2 + Zx 3 + 3.r 4 + • •
(c) cerca (le Ix' < 0.9
+
•
•
•
ix
2
—3
+
2x
—
—
—
(a) y =
(a) y i (x) = 1 — .1(x — 1) 3 — (x —1) 4 + (x — 1) 6 + • • •
Y2(x) = (x — 1) — 1(x —1) 4 — .A(.v — 1) 5 +( .r — 1) 7 + • • •
ti
X(x — 4) .4
A(?. — 4)(A — 8) 6
21. (a) y1(x ) = 1 — — x 2 +
2 !
4!
61
- 2)(X — 6)(X — 10) 7
— 2(X — 2)(X — 6)
+•••
+
y 2 (x) = x
x5
5!
. 7!
3!
1, x, 1 — 2x 2 , x — 3x 3 , 1 — 4x 2 + x 1 ..v — 1,r 3 +
1, 2x, 4x 2 — 2, 8x 3 — 12x. 16x4 — 48.v 2 + 12, 32x 5 — 160x 3 + 120x
22. (b) y = x — .v 3 /6 + • ••
Seciio 5.3
I. 0"(0) = —1,
0"(0) = 0,
0"(1) = 0,
0"(0) = 0,
p =
7. p = 1,
p = 00
p=
z:/,'"(0) = 0,
0"'(0) = —2,
0"'(1) = —6,
0"'(0) = —a0,
014)(0) = 3
(4) (0) = 0
(4) (1) = 42
0 14) (0) = —4a1
6. p = 1,
S. p = I
p= 3.
p=
1
9.
RESPOSTAS DOS FROBLEMAS
(c) p = oo
p = 00
(b) p = cc
(d) p = co
(e) p = 1
(h) p = 1
(g) p = oc
(0 p =
(i) P = 1
(1) P = 2
(m) p= oo
(n) p= co
( k ) P = 0
( I ) P = (22
a 2
a 2) a 2
(42 a 2)(22 a 2) a 6
x4
•rz 210. (a) yi (x) = 1
4!
6!
2!
[(2ni - 2)2 - a 2] (22 a2)a2
(2m)!
1 - a' 3 (32 - a2 )(1 - a2)
Y2(x ) = x 4-- 3! x- +
5!
[(2m - 1) 2 - 4 2 1 • • • (1 - a2)
(2nz + 1)!
yi (x) ouy2 (x) termina corn x" dependendo se a = tz è par ou impar
n = 0, y = 1; n = y = x: tz = 2. y = I - 2x 2 : n = 3, y =x - 1X3
ffix5 "ki x6 + • • • P (x) x kr4 ffix6TclCi x7 + • • •
11 ' VI (x) =
p = oo
- 4 .1( 5 + • • • , y 2 (x) = - it,x4 +
- 4x 6 + • • •
y i (x) = 1 - x3 +
(a)
-
4
p = 00
+ 45x 6 + • • • , y2 (x) = x +
+
y i (x)= 1 +'x
p = 7r/2
+ kris + 40-x7 + • • • ,
+ • • • . y2(X) = X - ix 317X5 • • •
y i (x) = 1 + h.Y3 + 12 x4 -
P= 1
Niio é possivel especificar condicöcs iniciais arbitrarias em .v = 0; logo, .v = 0 c urn ponto singular.
xn
x 2
6.y=l+x+- 4••••+-+ ••=e`
2!
n!
x 2
x4
X6
+ + y= 1 + - +
+ 2" • n!
2 2 • 4 2 • 4 • 6
y = +x +
+
+•••
19. y = I + x + x2 + • • • + x" +•=
1
I-x
,
xn
v3 r'
x"
20. y = a„ ( I + x + ,•-7 + • • • + ni + • • .) + 2 ( v+ :,i-,- .,- • • • +
n!
•2
= cc' - 2 - 2x - x=
= me + 2( e' - I - x 2
r6,
Y 2
X4
(- I ),,x2"
21. y = ao( l - .--- + 2-- - 2 - + • • +
+•)
2'n!
2 22 2! 233!
A.3
.v 2
.r5
X4
+ (x + 2- - 3- - 2 -4 + 3 • 5 + • • •)
= aoc - '
+ (x + 2:).
x3
3 - 2 . 4 + 3A-5• 5 + . ' )
1 3 + 4x'
1. I - 3x2 , 1 - lOx2 + 3x4 ; X, x - 3x 3 , X - 3x
(a) 1, x, (3x2 - 1)/2. (5x3 - 3x)/2. (35x4 - 30x2 + 31/8, (63x 5 - 70x3 + I5x)/8
(c) P I . 0; P2, ±0,57735; P3, 0. ±0,77460; P4. ±0,33998. ±0,86114;
P5 , 0, ±0,53847, +0,90618
/2
-
\\
Secijo 5.4
y=c i lx +11 -112 + C2IX + 11-3'2
1. y = ci x I c2x -22.
4. y = c i x - ' cos(2 In lx1) + c2x - ' sen(2 In lx1
3. y = c 1 x 2 + c2 x 2 In Ix'
6. y = c i (x - 0 43 C 2 (X - 1)-4
5. y = cl x + c2x InIx1
c.214-5-iftzba
y =
y = c1 1X1 312 COS(1 0 In Ix') + c2 1x1 2 sen (1./3' In lx1)
y = clx3 + C 2 X3 In I•I
y = c i (x - 2) -2 cos(2 In Ix - 21) + c 2 (x - 2) -2 sen(21nIx - 21)
y = C I I X I -1/2 cos(2 f1.3 In l x 1) + c21x1""2 sen ( 1,- 43. In 1x1)
y = cl x + c2x413.
y= 2x32 - x-'
14. y = 2x- "cos(2 In - x -112 sen (2 In x) 15. y 2x2 - 7x2 In Ix!
17. x = 0. regular
16. y = x- ' cos(2 In 19. x = 0. irregular; x = 1, regular
18. x = 0, regular; x = 1, irregular
571
572
5.
21. x = 1, regular; x = –1, irregular
23. x = –3, regular
25. x 1. regular; x = –2, irregular
27. x = 1. –2, regular
29. x = 0. irregular
31. x = 0. regular
33. .1 = 0. ±ru , regular
35. a < 1
37. y = 2
20. x = 0, irregular; x ±1, regular
22. x = 0, regular 24. x = 0. –1, regular; .v = 1, irre g ular
26. x = 0, 3, regular 28. .1 = 0, regular 30. x = 0, regular 32. x = 0, ±mr, regular
34. x = 0, irregular; x ±rur, re gular
36. fi >0
a>1
(a) a < 1 e > 0
a < 1eJ3> 0, ou a=lefl>0
a> 1 efl > 0
a > e /3 > 0, ou = 1 e > 0
>0
a=
41. y = ao (1 –
x-
2 • 55 + 2 . 4•5 • 9
44. Porto singular irregular
46. Ponto singular regular
48. Ponto singular irregular
45. Porto singular regular
47. Porto singular irregular
49. Ponto singular irregular
Se(*) 5.5
an-2
I. (1) r(2r – 1) = 0: a„ = (n + r)12(a + r) –11
.v 4
2 • 4 • 5 • 9
v.=
x ' 2 [1 —
•
r2 = 0
r1 =
X6
2 . 4 . 6 5 • 9 • 13 1."
12"n!S • 9 • 13 . • • (4n + 1) +
.v4
+
y2(x) = 1 –
2 • 3 2 • 4 • 3 -7 2 4 6 . 3 . 7 . 11 ±
(-1),:x2"
+ 2"n!3 • 7 ll• • (4n – 1)
an_2
ri = 5 , r2 =
2. (b) r2 – = 0; an =
(n + r) 2 –
+
r \ 2 ±
I
(c) y 1(x) = x 113 [ 1
1:1l+3)
1
1/2 )2!(1+:1i)(2-r
(-1)"
m!(1 -43)(2+
r
y 2 (x) = x -I/3 [1
1!(1
0 • • • ( in
+
1)
(2
1
tx \ 2 +
— 1)
1/2 )2!(1 – 0(2
(–iv,
7)7!(1i
– 5 )(2
(X 4
1
. ) k2
– 1
t x \ 2,n
– )• • • (In — 1) 1/2. )
+•
Sugesulo: faca a = 2m na relacilo de recorracia, m = 1.2, 3, ...
an_i
3. (b) r(r – 1) = 0; a„ = ; r1 = 1, r2 = 0
(a + r)(rz + r – 1)
(c)
x
Y1(x) =
4.(b) r2 = 0;
x`
(-1)"
'x. [ 1 – —
1!2! -4- 2—
!3 ! + * + n!(n + 1)!
a„ = a„_1
.
(a ± 62'
r i = r, = 0
x2
(b)
x
C"
y i (x) = 1 ± ow + (2!)2
±
— ± + _
a n(n!)2
r(3r – 1) = 0; a„ = – (n + r)[3(tz + r) – 11:
= r2 = 0
(c) Yi(x) = xi "
1!7
2
+
2!7 1 13 ( x2') ) 2
x2
(-1)'"
ni!7 • 13•• • (6m + 1) (
2)
+•
•
m
+•
"]
2
(d) y2 (x) = 1 -
115
(
(— 1)"'
) + 1 ( x 2 )2
2
2!5 • 11 2
nz!5 • 11 • (6m - 1)
na relacao de recorrencia,m = 1. 2, 3, ...
Sagestiio: faca n =
6. (b) r 2 - 2 = 0; a„
- y 1 (x) = x 12 [1 a
(n + r) 2 -
1(1 + 2 V2)
2
+
r1
q"'
r2 =
.v2
+
2!(1 + 2 vi )(2 + 2v2)
(-1)"
x" + • • -]
n!(1 + 24)(2 + 2 i2) • • • (n + 2.4)
x2
1(1 - 2./) + 2!(I - 2./72)(2 (-1)"
n!(1 - 2./724(2 - 24) • • • (n - 2,./2)
7. (b) 1.2 = 0; (n + r)a„ =
ri = r2 = 0
x2 x3
(c) y i (x) = 1 + x +
-3--,
• • + —
• = e`
n,
8. (h) 2r2 + r - 1 =
(2n +2r - 1)(n r + 1 kl„ + 2a,,_2 = 0:
=
r2
=
-I
r1
Y2
(-1)"'x'"'
(c) y i (x) = x 112 (1
7 217 • 11
ne7 • 11 • • • (4m + 3)
y2(x) =
[1
(d) y2 (x) = x - ' (1 - X 2 + X4
2!5 -.
. .+ + • • .)
( -1)mx2'
m!5 . 9 ... (4m - 3) ± )
9. (b) r2 - 4r + 3 = 0; (n + r - 3)(n + r - 1)a, - (n + r - 2)a„_ 1 = 0: r, = 3, r2
2
2x"
- x +—
v + +
(c) y l (x) = x' (1 + 2.
3
n!(n + 2) + )
(b) r 2 - r +
0; (n + r - ) 2a. ± a n 2 =
= r, = 1/2
_
r2
Z +
•+ (c) y1 (x) = x 1/2 (1 -—
22 4 2
22"'(m!)2
(a) r 2 = 0; r 1 = 0, r2 = 0
(b) y i (x) = 1 +
_14_1 y
CY(a + 1)
(x
1)
a(a + 1)11 • 2 - a(cr + I)]
1) 2
(x
2 12
22)
(2 1 2 )(2 •
a(a + 1)11 . 2 - a(a + 1)1— [n(n - 1) - a(a + 1)1
(x 1)"
2"(n!):
+ • • •
12. (a) r, = r2 = 0 em ambos x = ±1
(b) Y I (x) = Ix - 111/2
(-1)"(1 + 2a) . • • (2n - 1 + 2a)(1 - 2a) . • • (2n - 1 - 2a)
x [1 + E
(x
2"(2n + 1)!
ti= I
1)"1
Y2(x ) = 1
+
(-1)"a(1
+ a) • • • (n - 1 + a)(-a)(1 - a) • • • (n - 1 - a)
(x - 1)"
n!1 • 3 • 5•• • (2n - 1)
- ).)an-:
0, r2 = 0; a„ = (n - 1 n2
(—A)(1 — A.)- • • (n - 1 - A.)
(-A)(1 - A) 2
x + +
(c) y1(x) = 1 + (p )2 x +
(2!)2 (n!)2
13. (b) r 2 = 0; r1
Para A = n, os coeficientes de todos os termos depois de x" sao nulos.
16. (c) [(n - 1)2 - 1]!)„= -b„_ 2 e e impossivel determinar b,.
=
1
573
Secäo 5.6
(b) r(r - I) = 0; r1 = 1, r, = 0
(a) x = 0;
r2 - 3r +2 = 0; r i2, r2 = I
(a) x = 0;
(h) r(r - 1) =0; r 1 = 1. r2 = 0
3. (a) x = 0;
(b) r(r + 5) = 0: r 1 = 0, r, = -5
(a) .v = I;
Nab tern ponto singular regular
(a) x = 0;
L=.- -2,73
(b) T 2 +2r - 2 = 0; r1 = -1+ .7=3 0.732. r, = -I -
0; r: = r2 = 0
(b)r(r -
(a ).v = 0:
(b) r(r - = 0; r1 = r2 = 0
(a) x = -2;
(b) r2 + 1 0: r t = r, r, = -1
(a) x = 0:
S. (a) x = -1;
(h) T 2 -7r t- 3 --= 0; r1 = (7 + ./37)/2 -1' 6,54. r2 = (7 - 41)/2 1-1' 0.459
(b) r2 + r = 0; r 1 = 0, r2 = -I
9 (a) x = 1;
(b) r 2 - (5/4)r = 0: r l =5/4, r2 = 0
(a) x = -2:
= 2, r2 = 0
(b) r 2 -2r = 0;
(a) x=2:
(b) r 2 - 2r = 0; r1 = 2, r, = 0
(a) x = -2;
(b) r 2 - (5/3)r = 0; r 1 = 5/3. r2 = 0
(a) x = 0;
(b) r 2 - (r/.3) - 1 = 0: r 1 = ( 1 + .137)/6 "L- 1,18,
(a) x = -3;
r, = (I - .,71)/6 -0,847
(b) = 0, r2 =00
(c) y 1(x) = 1+ x + yx 2 + i; •3 + • • •
+••.
-
y, (x) =3 7 1(x) In x — 2x -
(b) r 1 = 1, r 2 = 0
•••
- N X 4
(c) y i (x) = x - 4x 2 +
-6y, (x) In x + I - 33x 2 + "Tx3 + • • •
Y2(X)
(h) r 1 = 1, r, =0
(c) Y1(X ) = -t-
+ 1X 3
+•••
2. 2 - Vx 3 + •
y 2 (x) = 3y, (x) In .v + 1 - 4x
(b) r 1 = 1, r2 = 0
1 4 ,
,2
( c ) y i (x) = x
144"
12'
+
-
+
Y2( X) = -Yi (x ) In .v + I -
(b) = 1, r2 = -I
( C ) Yi(x)=x-1 3+
(x) In x +
- 4.v3 + • • •
Y2(x) =
18. (b) r 1 = r2 = 0
(c) yi (x) = (x -1) 1/2 [1 - i(x -1)+ (x — 1) 2 + - .1. (d) = I
19. (c) Sugestilo: (n -1)(n - 2) + (1 +a + f3)(n - 1)+0 = (n - 1 + a)(n - I + /1)
(d) Sugestlio: (n - y)(n -1- y)+ (1 + + 01(1- y)±0(13 = ( n - y +a)(11- y +13)
Seca() 5.7
cc
1. yi(x)=
E
n=0
(_1)"x"
n!(n+1)!
ii„ +
II„
In x x+ [1 - E (
n!(n - 1)! 1)".0
1 °`' (-1)nx"
2N-,
-"‘" (-1 ("11„n
y i (x) = - E
y 2 (.v) = (x ) In x
x n=o (n!)2
(n!)2
n=1
Y2(X ) =
yi
(x)
(-1)"2"
=
n=0
(n!)2
Y2(x ) =
(x) In x - 2
- (_i)"2"H„
E
1,1
1
xn
E
4. y i (x)= x- n_o
n!(n+1)!
Y2(x) =
(x) I n x + 1
x2
11„+11„_1
— 1)!
)".e
(n!) 2
(_ 1.)m
5. YI(x) == A3I2 [1 „,. 1 101 +
-V-312[ 1 +
Y2( X ) =
\
N2+ 0 • • • (In +
E m!(1 - 4)(2 - ) • • • (1,1 -
(:_x• \ 21,1]
)
k 2)
Sugesain: faca n = 2nz na relacäo de recorrencia. III = 1, 2, 3....
Para r = -4, a, = 0 e a, ë arbitrzirio.
CAPiTCLO 6
Seciio 6.1
I. Seccionalmente continua
3. Continua
(a) F(s) = 1/.5 2 , s >0
(c) F(s) = tz!/sn+1, s > 0
F(s) = s gs2 a2), s > 0
8. F (s) =
b
14. F(s) = s a > Ibl
(s - a) 2 - h2
12. F(s) = s2 s
b2 ,
S-
a
s>a
(s - a)2+ b'
2as
, „ s > 0
(s- + a- )n!
18. I; ( S) = S>a
on+,,
a2)
20. F (s) =2a(3s2
> lal
(s2 - a2)3
16. F(s) = ,
I 11/2) =
/2:
11.
(s)
b: .
s > Ibl
S- a
s-a>
(s - a) 2 - G2'
s2 + 1)1' s >
Ibl
0
h
s>a
(s) = (s - a)' + b2'
1
S>a
15. F(s) =
,
(s - a)s' + a17. F(s) =
s>
- a) 2 (s+ (1)2'
2a(3s : - )
s>0
19. F(s) = ;
(s- +
13. 12
s>0
21. Converge
23. Diverge
26. (d) 1(3/2) =
7. F(s) = s2
9. F(s) =
s>
s2 -b- b2
10. F(s) = 2. Nenhuma das duns
4. Seccionalmente continua
(b) F(s) = 2 s3s > 0
22. Converge
24. Converge
32
Secao 6.2
1. f (t) = ; sen 2t
3. f(t) = e t ie-41
5. f (t) = 2e" cos 2t
7. f (t) = 2e.' cos t + 3e' sen
9. f (t) = -2e -2( cos t + 5e -:' sen t
11. y :(e3r + 4e-2`)
13. y = el set) t
15. y= 2e' cos
t - (2/ ifl)et sen ./5
17. y = te` - t 2 e: + 313e'
19. .v = cos Nif t
21. y= (cos t - 2 sen
4e: cos t -2e`sent)
23. y = 2e -' te" + 212e-1
25. Y (s) =
1
s 2 (s2 +
e-` (s + 1)
1)
2. f (t) = 2t2e'
4.
f (t) = e3' +
6. f (t) = 2 cosh 2t - sehn 2t
8. f(t) = 3 - 2 sen 2t + 5 cos 2t
sen 3t
10. f (t) = 2e' cos 3t -
- e-2:
12. y =
14.
y = e:r - te2'
16. y = 2e -t cos 2t + ; e - ` sen 2t
18. y = cosh t
20. y= )w2 - 4) - '[(w2 - 5) cos cot + cos 2t]
22. y= i(e- ` - e2 cos t + 7e sen t)
1S
24. Y(s) - + s2 + 4 s(s2 + 4)
26. Y(s) = (1 - e- c)/s2(s2 + 4)
s2 (s: + 1)
30. F(s) = 2b(3s2 - b2)/(s2 + b2)3
29. F(s) = 1/(s - a)2
32. F(s) = n!/ (s - art
31. F(s) = rz!/ sn+1
F(s) = [(s - a) 2 - h'- 1/[(s - a)- + b2]2
J2
+
b2
3
4.
F(s)
=
2b(s
a)/[(s
a)
:
33.
(b) s2 Y" + 2sY' - [s2 + a (a + 1)1Y = -1
36. (a) Y' + s2 Y = s
Seciio 6.3
(b) f (t) = -2u3 (t) + 4u5 (t) - u7(t)
(b) f (t) = 1 - 2uz (1) + 2u 2 (t) - 2u3 (t) + u4(t)
('
10. (b) f (t) = r 2 + u 2 (t)(1 - 12)
9. (h) f( t ) = 1 + 11 2 (0(e - -2) - 1]
2)
(t)(t
11
3
(b) f (t) = t - u 1 (t) - u2 (t) (b) f (t) t + 1(2(0( 2 - t) + 11 5 (0(5 - t) - u 7 (t)(7 14. F(s) = e -s (s2 + 2)/s3
13. F(s) = 2e -s 1 s'
e-2"
e -"
16. F(s) = 1 (e' + 2e -3s - 6e-4s)
(1 + 7s)
15. F(s)
-s2 - -s218. F(s) = (1 - e-s)/s2
17. F(s) = S 2 ((1 - s)e -24 - (1 + s)C3s]
f (t) = 102 (0[e t - 2e-2(1-211
t3
e2'
20.
19. f (t) =
22. f (t) 11 2 (r) senh 2(t - 2)
21. f (t) = 2u 2 (t)e' cos(t - 2)
24. f (t) tt i (t) + "2 (t) - 11 3 (0 - 114(t)
23. f (t) = it i (t)e2(1-1) cosh(t - 1)
cos t
27. f(t) =
26. f = 2(2t)"
t • 112(t 12)
f(t)
=
29.
28. f(t) = e'13 (013 - 1)
s>0
31. F(s) = s -1 (1 - e -' + e-2s -
30. F(s) = s- 1 (1 - Cs ), s > 0
e-(2n-,21.5
1
1
s>0
32. F(s) = - [1 - e S + • • • + e-2"s - e-(2n+11
s(1 + e -s ) •
(-1)"
-
33. F(s)
.
1 / s
1 + cs
n=0
35. 4f
- 1/s
s>0
,s> 0
1 + e-s
1 - (1 +s)e-`
- sz(1 _ e-s) , s > 0
37. Cif
39. (a) ,C{ (0) = s- 1 (1 - e -5 ). s > 0
r(g(t)) = S -2 (1 - e'), s > U
.C{h(t)} = S -2 (1 - e - ') 2 , s > 0
- e-5
s>0
40. (b) .4)(0) = s2 (1 + e-5)
1 - e-s
s>0
5(1 + e-5)
1 + C"
38. r(f (0) =
1 + s2 )(1 - e- T '
36. EV
=
s>0
Sectio 6.4
(a) y = 1 - cos t sen t - u 3T (t)(1 + cos t)
Sell ti
(a) y = e-' sen t + lu,(t)(1 + e - " -') cos t +
-tt2„(t)rt - e-(' -2 " ) cos t - e-"-'-"sent
(a) y = [1 - u 2 , (t)](2 sen t - sen 21)
(a) y (I? (2 sent - sen 20 - tt.,(t)(2 sen t + sen 20
- e u-IN
+ le -21 - e-' - t€ 10 (011,
(a) y
+
;e-2"
'
e
-u
-2
u
+
2
(t)(1,
e-21
y
=
e-'
(a)
(t)(1 - cos(t - 37)1
(a) y = cos t +
(a) y = h(t) - tt, 12 (t)h(t - r/2). 11(t) = (-4 + St + 4e -'t2 cos t - 3c o sen 0
sen t + - ite,(0[t - 6 - sen(t - 6)]
(a) y
(a) y = h(t) + it, (t)h(t - 7), li(t)= (-4 cos t + sen t + 4e-o cos t + co sen t I
)1
- cos(2t -
(a) y = 11,(t)[; - cos(21 - )] - 113,
(a) y = 11 1 (t)h(t - 1) - 112(1)/1(1 - 2), h(t) = -1 + (cos t + cosh 0/2
(a) y = h(t) - u_(t)h(t - 7), h(t) = (3 - 4 cos t + cos 20/12
f (r) = (110)(t - to) - 11 zo+k (1)(t - to - k)1(11/ k)
g(t) = (1110 (t)(t - to) - 21110 +k (0(1 - to - k) + u,, !-2k ( 1 ) (t - to -
2k)](h/ k)
- 4k11 512 (t)11(1 - 1),
h(t) = - ( 7/84) e- lis sen(30 t/8) - co cos(3 ‘ 17 t/8)
(d) k = 2,51
(e) r = 25,6773
(a) k 5
(b) y [us(t)h(t - 5) - u 5+1,(t)h(t - 5 - k ))1 k, h(t) = - a sen 2t
(b) fk (t) (tr 4 _k (t) - 4+k(t)112k;
y = Itt.t_k(t)h(t - 4 + k) - 114 k( t ) I1 ( t - 4 - k))12k
(b) u(t) = 4ktt3112(t)h(t -
h(t) =
a -
IC` i6 cos( 143 t/6) - (.543/572) e -06 sen(,/iT3 t/6)
19. (b) y = 1 - cos t + 2
E (- o k u , (t)i 1 - cos(( - k7))
k
k=l
21. (b) y
1 - cos t + E(-1)k uk, (I) (I - cos(t -
)1
k=1
!I
23. (a) y
1 - cos t + 2 E (-1) k uilk/4(o[ - cos(t - tik/4)]
k =1
6.5
(a) y = c' cos t + c t sen t + u, (t)e-(' ) sen (1 - jr)
(a) y= Itt,(t)sen2(t - 7) - Itt,„(t) sen2(t - 27)
+ 1110(0 [1 + e-2(t--10)
(a) y =
+
+ 115(t)[-e-20-5, e -u
(a) y = cosh(t) 2044 3 (0 senh (t - 3)
(a) y = sen t - i cos t + le' cos Nr2 t + (11 ./.) u 3, (t)e-(1-3T) sen 12- (t - 37)
(a) y = z cos 2t + tt 4n (t) scn 2(t - 47)
(a) y -= scn t + u2,(t) sen (t - 27)
(a) y = u, f4 (t)scr12(t - 7r/4)
(a) y = ti, /2 (0[1 - cos(t - 7r/ 2 )]+ 311370(t)sen(t - 37/2) - il2,(1)[1 - cos(t - 27r)]
(a) y = ( 1/./f) 11, 16 (t) exp[ - (t - 7/6)]sen(OT/4)(t - 7/6)
'sen t +it,i2 (t)e - (4 -N /2) sen (t - n/2)
11. (a) y = 31 cos t + sen t - -5-e"
cost -
1 cos
P. (a) y = u l (t)(senh(t - 1) -sen([ - 1))/2
(a) -e-274 8(t - 5 - T), T = 87/
(a) y = (4/./7175) tt 1 (t)e-"-11t4 scn(./75/4)(t - 1)
t1 L 2,3613, y ) .1= 0,71153
y = (8./7/21) u i (t)e-"-10 sen (3018) (t - 1); t i L 2,4569, y L L 0,83351
(d) = 1 -F 7r/2 -24 2,5708, y i = 1
(c) k 1 = 2
(a) kl IL 2,8108
(b) k 1 L 2,3995
(a) 0(t, k) = Ett,s_k (t)h(t - 4 + k) - 114.14(01(t - 4 - k))/2k, h(t) = 1 - cost
(c) Sim
(b) 0„(t) = u4 (t) sen(t - 4)
Secau
20
20
IS. (b) y = E(-1)" I li k,(i)sen(1 - k7r)
17. (b) y = E uk,(t)sen(t - k7r)
k=1
20
k=1
20
19. (b) y = E uk, a(t)sen (t - k7r /2)
20. (b) y = E(-1) " I tik,12 (t)sen(t - k7/2)
14=1
k=1
15
40
21. ( b) y = E u(2k_ i),(t)sen[t - (2k - 1)7] 22. (b) y = E(-0kf.,,,,,i4(t)scn(t - 11k/4)
k=1
k=1
7.. 20
(b) y= *59 (-1)"1/4,(t)e-(1-icro0senk/399(t - k7)/20)
k=i
15
(b) V
20
= - E 11 (2k _
.1399
(
t)e- [4- (2k - t).,41/20 sen{st - (2k - 1)7)/20)
k=1
Secão 6.6
sen t * sen t = ,; ( scn t - t cos t) e negativo quando t = 27, por exemplo.
5. F(s) = 1/(s + 1)(s2 + 1)
F(s) = 2/s2 (s2 + 4)
7. F(s) = s/(s2 + 1)2
6. F(s) = 1/s2 (s - 1)
8. f (t) =
10. f(t)
(t -
f
= f
P. (c)
r) sen r d r
9. f(t)
11. f(t) =
(t - r)e-(` - `) sen 2r dr
um (1 - u)" du =
1)1(n +
1(m + n + 2)
(In +
=f
f
e-(1-T) cos 2r dr
sen([ - r)g(r) dr
1)
I
i
1
e' -`) sen(t - r) sen a r dr
13. y= - sen (ot + - I senw(t - r)g(r) dr 14. v=
to
co 0
o
y = y
f e -(`- `)/2 sen 2(t - r)g(r) dr
0
r
v = e"12 cos t - le-02 sen t +
C(`- ')/2 sen(t - r)[l. - u, (0] dr
o
4
v = 2e-2' + 1e-2r + (t - r)e-2('-')g(r) dr
f
v = 2e-' - e -2' +
f'
- e-2('- ' ) ] cos ar dr
1
19. y = - [senh(t - r) - sen(t - rflg(r) dr
'
2 0
578 RESPOSTAS DOS FROBLEMAS
7.
y = ; cost - 3 cos 2t + f[2 sen(t - r) -sen2(t - r)Jg(r) dr
F (s)
(13(s) =
1 + K(s)
(a) 0(t) = (4 sen 2t - 2 sen t)
(a) 0(t) = cos t
0'(3) = 0
0(0) = 1,
(b) 0"(0+ 0(0 = 0,
(a) 0(t) = cosh(t)
0'(0) = 0
0(0) = 1,
(b) 0"(t ) - 0(t) = 0,
(a) 0(t) = (1 - 2t + t2)e-g
0(0) = 1,
0"(0) = -3
(b) 0"( t ) + 20' (t) + 0(t) =
sen(Ot/2)
cos(,75t/2)
+
1
3.
e
/2
- e l2
(a) 0 (t) =
0(0) = 0,
0'(0) = 0,
0"(0) = 1
0,
(t)+0(t)=
0
(b)
(a) 0(0 = cos t
0'(0) = 0,
0(0) = 1,
0"(0) = -1,
(b) 0( 4 )(t)- 0(t) = 0,
28. (a) 0(t) = 1 - .e-1/2sen(0/12)
(b) 0'(t) + 0"(t) + 0'(t) = 0,
CAPiTUL 0 7
co) = 1,
o'(o) = -1,
0"'(0) = 0
0"(0) = 1
Seca() 7.1
= x_,
2.
'2 = -2x, - 0,5x 2 + 3 sen t
-2x, - 0,5x2
3. x'1 x2 , x'2 = -(1 - 0,25t -2 )x 1 - t -l x24. x1 = x 2 , x'2 = x3 , x3 = x4 , x4 = x,
x = x2, x'2 = -4x 1 - 0,25x2 + 2 cos 3t, x, (0) = 1. x2 (0) = -2
= x2 , x'2 = - q(t)x 1 - p(t)x 2 + g(t); x1 (0) = u0 . x2 (0) = tt'o
(a) x 1 =
+ c2e -3`, x2 = c l e' - c2e-3'
c, = 5/2, c 2 = -1/2 na solucâo em (a)
0 grafico se aproxima da origem no primeiro quadrante tangente a reta x, x,.
8. (a) x'1' - xi - 2x1 = 0
(h) x= 11-e2' - ie1 , x 2 = e2' - le'
(c) 0 grafico c assintOtico a reta x, = 2x, no primeiro quadrante.
9. (a) 24 5x', + 2x, = 0
e2r
x, = - ie/2 _ l e2f , .x2 =
0 grafico a assintOtico a reta x, = x2 no terceiro quadrante.
10. (a) xi + 3x1 + 2x, = 0
18.
1. x', = x2 ,
x'2 =
x 1 = -7e-` + 6e 2r , x 2 = -7e'` + 9e'
0 grafico se aproxima da origem no terceiro quadrante tangente a reta x, = x2.
11. (a) x',' + 4x, = 0
x1 = 3 cos 2t + 4 sen 2t, x 2 = -3 sen 2t + 4 cos 2t
0 grafico 6 urn circulo centrado na origem corn raio 5 percorrido no sentido horario.
12. (a) x',' + x', + 4.25x, = 0
xl = -2e -'12 cos 2t + 2e -o sen 2t, x2 = 2e-'12 cos 2t + 2e-ra sen 2t
0 grafico a urna espiral se aproximando da origem no sentido horario.
13. LRCI" + LI' + RI = 0
_VI = y3,
= Y4, tn i y; = -( k 1 + k2)yi + k2y2 + Fl(t),
rn2y4 = k2y1 - (k2 + k3)y2 + F2(t)
22. (a) Qi = i - -,1,5 Q, + Q2, Qi (0) = 25
(2'2 = 3 + ;WI - 5Q2, Q2(0) = 15
Qi = 42, Qi =36
= qdx, + x2 , x 1 (0) = -17
- 1X2 , x 2( 0) = -21
x2 =
23. (a) Qi = 3t/i - is Qt + yiro Q2, Qi (0) =
Q'2 = (12 + 3a Qt —10o Q2, Q2( 0 ) =
Qi = 6(9q1 + (72), Qf = 20(3q 1 + 2q2)
Não
(d) 192 < QPQ <
Seciio 7.2
6
—6
9
3
1. (a) (5
2
6
4
(c) ( 9
3
—2
8
—12
3
12
(b)
—15
7
—26
6
—18
—3
3
—8
—9
7
(d)
14
12
0
5
—8
—7 + 2i) th \ ( 3 + 4i
2 + 3i) ' I 11 + 6i
2. (a) ( 1 — i
—1 +2i
7 + 5i)
(-3 + 5i
(d) (8 + 7i
6 — 4i
—2
1
2)
1
3
3. (a) ( 1
0
—1
(b) 2
—1
2
—3
3
1
—1
—1
4
0)
(c). (d) ( 3
— l
0
5
—4
1
(c)
2+i
7+2i
2i
1 + i
2 — i) (b) (3 + 2i
2 + i
—2 + 3i
3-
4. (a)
(10
5
0
4
6
4
4
—12
—1
—5
11
—5
5
6i )
6 — 50
4 — 4i
—4)
—2
1
0
1—i)
—2 — 3i)
—1
4
4)
7
1
(c)
(b) 12 — 8i
( 2s_
(d) 16
1
1
—3
2
li
11. (
1
11
(
P.
(c) 2 + 2i
..._ 1.)
11
10.
—2
(
2
—3
3
—1
—1
0
13.
_
1
3
3
1
3
15.
3
10
4
10
_ 1
10
16.. (-1
10
_ 7
10
/ 1
18.
0
21. (a)
(c)
(7e
—et
8e'
et
2e'
—e
0)
—1
1
3
0
3
(1
—s1
0
1
0
0
2
14. Singular
1
10
2+i
—2 — 3i)
—14(0)
7
—11
—3)
5
6. (a) (0
11
20
17
(h)
2
—4
3
—12
—1
6
—8
—11
9
15
6
—1
5
( —5
8. (a) 4i
(c) (3 + 2i
'c' 1 — i
3
1)
8
_
4
1
2
1)
10
17. Singular
to
3
10
1
0
1
)
0
1
1
)
1
1
1
1
0
1
5e'
7e'
0
—2e-'
—3e
19.
6
y
_ 5i
5
11
-5—1
—3
0
2e2'
—2e2'
4e2'
1
5
4
—2
10e2' )
2e21(b)
—e2 r
6
5
4
3
1
5
_1
—5
5
5
2e21 — 2 + 3e3'
1 + 4e-2' — e'
4e" — 1 — 3e"
2 + 2e-' + et
—2e' — 3 + 6e'
—1 + 6e.- 2: — 2e!
(d) (e — 1)
1
2e-1
2
—1
3e -I
— ee ++ 11))
e +1
3e3' + 2e' — e4')
6e' + et + e'r
—3e3` + 3e' — 2e4'
580
RESFOSTAS DOS PROBLEMAS
Secao 7.3
1. x 1 =
7
X2 = 1 ,
2. Nâo tem solucao
X3 —
x i = — C, X2 = c 1, X3= c, onde c e arbitrario
x i = c, x 2 = —c, X3 = —c, onde c é arbitrario
6. x 1 = c 1 , x2 = c2, X3 =
5 - X 1 = 0, x, = 0, X3 = 0
8. x''' —5x( 2 ) +2x( 3 ) = 0
7. Linearmente independence
14. Linearmente independente
1
1)
= —3, x(1) —
;
A 1 = 0, x (1) =
16. A l = 2, x (1 ' = (13) ;
2i, x
A2 = 1 —
(2)
;1
A l = 1 + 2i, x(1) = ( 1 —
Al
10. Linearmente independente
13. 3x")(t) — 6,C 2) (t) + x (3) (t) = 0
=0
9. 2x ( ') —3x( 2 ) +4x( 3 ) —
I I. x") + x( 2 ) — x( 4 ) =0
;., 2
A2 =
A l = 2, x" ) = (
=
—2 . x121
A2 = 4,
x (2 ) = (11)
= ( +
1 i)
Ci )
=
— 1 , x(2 '
2, xa)
A2 =
2c2 + 2
1)
=
(
)
1
A l = —1/2, x( 1 ) = 3 ) • A 2 = —3/2 x (2) = ()
(I() '
(
0
= 1 — 2i, x( 3 ) = (1
i
0
2
1) :
A2 = 1+ 2i, x( 2 ) =
;‘, 1 = 1. x( 1 ) =
2
0)
1
23. A l = 1, x (1) =
A 2 = 2, x( 2 ) =
;
1) ;
0
( —1
2
24. A 1 = 1, x( 1 ) =
A3 =
3. x 13 ) = (
1
2
A2 =
-2) :
2, x( 2) =
_
(1) :
1, x(3) = (_)
21
( -1
1
25.
= —1, x (1) =
-4
;
A2
— 1, X' 2 ' =
1
0
—1
:
;+. 1
= 8,
7e 3 ) = (1)
2
2
Secäo 7.4
2. (c) W(t) = c exp
iPti(t ) +P22(01 dr
6. (a) W(t) = 12
Ye( ) e x (2) silo linearmente independentes em todos os pontos, exceto em t = 0; eles silo linearmente
independentes ern todos os intervalos.
Pelo menos um coeficiente tern clue ser descontfnuo em t = 0.
(0
1
(d) x =
—2/- 22r)) x
7. (a) W(t) = 1(1 — 2)es
x") e x( 2 ) sac) linearmente independentes em todos os pontos, exceto em t = 0 e t = 2; cies silo linearmente independentes cm todos os intervalos.
Pelo menos urn coeficiente tern clue ser descontinuo em t = 0 e em t = 2.
0
1
(d) x' = 2( — 2t
t2 — 2 x
t 2 — 2t
12 — 2t
SecSo 7.5
2
1. (a) x = c ) (21 )e" + c 2 (1)e 2t
?. (a) x = c i ( 11 ) e-: + c 2 (2)
3 e -2d
3. (a) x = (- I CI ) ei + c 2 (31 )e'
4. (a) x = ci ( -41 )e-31 ± C2( 1 ) e2t
5. (a) x = e l (2) e -3' + C2 CI ) e-t
6. (a) x = c 1 (_ 1i )e, + c2 ( i1 ) e-,,
8. (a) x =- c 1 (- 1 ) ÷ c2(-31)e`
7. (a) x = c 1 ( ) + c 22 (2) C -2(
4)
. C1 (2 i\ e; _ m i 1\ _„
1 + c 2 ( 1 ) eV
10. X
9. x = el()
c--A-lr
-1 )
i
-i
1
11. x - c i (1)
1
1
1
(-2)e` + c 3 ( 0) e'
-1
1
e4' +
2
1
0 e--` + c 3 1 e'l
'?
-1
1
(1 e,,
4 e_,, _r (.2 _ .4 c_, + c3
1
e -: + c 2
x = c1-(4
x
= c•;
__./
-7
1
14. x = c l -4 et
-1
1
-1
+ c2
c -21 + c3
-1
-1
15. x = - 3 (31 )
1
2
1
16.
e21 + -/ (I1) 4(
1
1
0
0
17. x= ( -2 e' +2
1
x = 11 11) e, + ,,1. 0 e3,
_W1
/e
18. x = 6
e2'
1
1
1 I t + -( 3 )t-1
20. x =
ear
21. x =
1
22. x = c l (3)
4 + c,- 2) t--,
(a) = x,, .e2 = -(c/a)x 1 - (b/a)xz
(1
2
-1
et
+ 3
-2 e -1 1
1
-8
( )12
3
23. x = c: ( 21 ))
(2)
t2
(a) x = 55 C)
2 e - ti.20
• + 29 1 ) e14
(c) 7'
74.39
e` -2+4)02
31. (a) x = c1
r1.2= (-2
C2
(4) e1-2-4)//2.
1
± 4)/2: no
x = c 1 ( -1)
e (-1+1211 + e 2
(.7
1
2)
r1.2 = - 1 ±
ponto de sela
= -1 ± „AZ, a = 1
32. (a) ( v )
= c l ( 31 )
e
-2' + C2 ( 1 ) e-t
33. (a) (
1
CR
Sectio 7.6
cos 2t
+sen2t
1. (a) x = clef ( cos 2t +sen 2t) + czet ( - cos 2/ sen 2/
4
2
L )
CL
0
eit
581
582
(a) x = cie" (
2 sen 2t
2 cos 2t
+ c2e-t —
cos 2t
sen 2t
5 sen t
(
5 cos t
+ C2
— cos t + 2 sent)
(a) x = (2 cos t + sent
it
(a) = cie'12 3(cos 5 cos+ sen it))
)
cos t
(a) x = cle " z cos t +sent
c 112 (
2e
+ QC' (
( —2 cos 31
(a) x = ci cos 3t + 3 sen3t -r
0
8. x c1 (-2)
1
sen t
—
cos t + 2sen t
—2 sen 3t
sen 3t — 3 cos 3t)
x = c l ( - 3) e t + c2 e (cos 2t) + c3e 1
sen 2t
0
sen 2t
— cos 2t
sen f t
cos f t
— cos 4 r — ,r2- sen t
P -2'
(cos t — 3 sen
COS —sent
9. x = e'
5 sen it
3( — COS i t + sen it)
+ c3e
f cos t
sen ./2 t
../1 cos 4 — sen t
10. x = e2t ( cos — 5 Sent
—2 cos t — 3sen t
12. (a) r =
i
11. (a) r =
(b) a = 0
(a) r a i
0, ./Y)
(b) a =
— 20)/2
(a) r = (a ±
16. (a) r = 1 ± 107e
15. (a) r = ± 14 — 5a
(b) a = 4/5
(b) a = 0, 25/12
(b) a = —1, 0
(a) r = —1+ V:-Te
(a) r =
(b) a = 2, 49/24
± 2./49 — 24a
(a) r
(b) = —4 — 21T), —4 + 211b, 5/2
a — 2 ± Va 2 + 8a — 24
(a) r = —1 ± 125 + 8a
(b) a = —25/8, —3
cos(41n t)sen(121n t)
21. x =
sen(4 In I)) C2t
COS(I In t))
(
5 sen(ln 0
5 cos(ln t)
"Y? X = C1
-F C,
2 cos(ln t) +sen(In t))
- (— cos(In t) + 2 ser(In t))
23. (a) r = — 1 ± i, -- .1
24. (a) r = — 1 ± i, to
( 1 = _ _ i n ( cos0/2) ) + . _ 02 ( sen(t/2) \
25.(n)
/ ) `le
4 sen(t/2)
c ' e
—4 cos(t/2))
Use c1 = 2, c2 = — i na resposta do item (b).
lim 1(t) = lim V (t) = 0; nao
I-
. :"C
1.-,•W
26. (b) ( v) = cie-'
cos t
(
— COSI — sen t
+ c,e"
sent
—sen t + cos t)
Use c 1 = 2 e c2 = 3 na resposta do item (b).
lira 1(t) = lim V (t) = 0; nä°
(b) r = ±i11717n
(c) ri = —1,
(d) Irl e a frequencia natural.
") =
23 ) ; r2
—4,
3
(2) = (—4
x l = 3c 1 cos t + 3c2 sen t + 3c3 cos 2t + 3c4 sen 2t,
x, = 2c 1 cos t 2c2 sen t — 4c3 cos 2t — 4c., sen2t
x'1 = —3c1 sen t + 3c2 cos t — 6c3 sen 2t + 6c4 cos 2t,
x'2 = — 2c 1 sen t 2c2 cos t + 8c3 sen 2t — 8c4 cos 2t
)
0
0
3
—13/4
0
30. (a) A = ( °4
9/4
1
0
0
0
0
1
0
0
1
(b) r1 = i.
11) = (
(3)
r3 = i= i,
i\
1. ) :
i
4\
—3 .
10i '
— 15
i t• I
(
(2) =
r2 = —i,
—i
—i )
4
r4 = —li ' (4) =
1-03'
—
( 15
2
/
( cos t \
sent
cos t
sent
+ C2 ( COS I
—sent
(c) y = el
sent j
cost
C2 = 0, C3 =
0
0
31. (a) A= (
—2
1
C4
01
0
\
/
—3 cos P
+c3+c4
—10sen .. t
\
(e) c 1 = 10
4 cos P
= 0.
15 sentt
7
/
4 sell; t \
—3 seri i t
10 cos
Zt
15 COS5 1 J
\—T
period° = 4n.
0)0(
1
01
0
1 \
=(
(b)
(2) =
r2
i
1
—1
(
r3 = 0i.
(3) —
r4 = — 0i,
A ) ;
(4) = (
— 1)
—vi
(c) y = c 1(
cos t \
cos t
/
cos t )—sent
sent J 4- c'2 costttt ± c3
(e) c, = 1. C2 = 0, C 3 = —2, c4 = 0.
Sec:10 7.7
1. (b) (I)(t)
3
=
3 e
2. (b) (NO = (
+ 3 e2'
2
5e-,
- I
`
li e-s/2 + -;e"
1 e -g/2 — l4 e-'
(b) OM .--- ( fr — C'
-i2 ef — 12 e-'
(b) ^(t) =
le-31 + ie2r
+ 16,2,
_
cos t + 2 seri t
5. (b) ^(t) =
sent
(b)
+
AL_
cl)(t) =
( e" cos 2t
sen 2t
2
1 e2' + e
(b) (NO =
2
2
2e4(
2
2
3e-r
_2 e2r )
3
—
3
13 e2t
e—t/2 —
el
le-0
+ 2Ict
2
—1e` + le'
— -21e i + 3-2 - I
15
-3t
-1 e 2t
—5 sen t
cos t — 2 sen
—2e' sen
e' cos 2t)
e2r
12 ea,
-2 2:— -21 e 4e
3e
cos 13 t \
— cos 0 I
± C4
—0- sen0 t
.173 sen13 t /
sen 13 t
sen 0 t
Nij COS N/7
3 t
— 0 cos 13 t
(—
2
e'sent
e -t cos t — 2e-`sent
(e-1 cos t + 2e' sen t
5e" sen r
8. (b) (1)(t)
—e-21 + e-1 ( —2e-2( + 3e -'
- r
- l t
5 —1312 e ,I
-4 e - F
-—,e
e
e + i2(
e -2i — 4-1
,4
(b) 4)(r) -=
74- e -2t — 27 e -t — , 3e-,t
e2c
Z e -2' — 2e-' — 22
1_
)
—)1 e' + 13 e-2t
ie.' — 1 e-2(
.,1 er — -31 e -2t
l6 e' + l3 e-2' + 12 e3'
(b) ^(t) = ( —der — le- ' + e3'
— 61er — 3-1 e--1 r + -21 e 3t
11. x = —7 ( I ) e` — 2 ( 3 ) e-t
1
17. (c) x
( /to ) cos wt +
vo
_e-2t
-1
+ le
12 2'
2.e21
—
-.
1
—
ie
ie-21
1 t -2t — 1 e31
-e —e
?:
4- e -D — .4 e'
+2 e3(
+ e-21—..-21e-11
—2e' + e-21
1 t
— e
3
12. x = ( ) e-' cos 2t +
c'sen2t
sen wt
vo
—(02110
Seca() 7.8
(1) et + c 2 [(I) te t + (0) et]
(c) x =
(c) x = ci (,)) + c 2 [0) r — (?)]
(c) x= c 1 ( I ) e-r + 2 R I )
+ (2) e-]
0 _, ,..,
,
1
(c) x = c i ( i ) e-1/2 + c 2 [( 1 ) te ( 1 - + () e ]
1
1 ( 01
• ( 01
—3
re' + 0) e2'
e2' + c.
( e -' + c2
5. x = c i4
1
—1
—1
_
((1
1
1
1 e-`
0)e -1 + c 3
x = ci1 e 2' + c 2
--1
—1
1
(a) x = (32+-I-4t4t) e
8. (a) x=
10. (a)
I te(12
9. (a) x = —2 )ert 2 +
2 ( -1 )
11. (a) x =
0)
(0
( —1)
24
1 te' + 3 0 e2'
+
—6
2
4 1
(a) x =__ (i) e - ' /2 + 71' ( 5) e-71/2
3 1
—7
°
x = ( I ) t + c2 [( 1t In t + ( 0 ) t]
14. x =
( I1 )
16. (b) ( 1,) = —
1
+ c-,[()
- 1 t-- Int
0
(I)t
2
1
) e-r'2 + [(_ 2 ) to -112 + (0) Cr'
= 2
3) e — 6 (11)
+ 14
3i ) r
(0
1
—1
( 0
1
(c) x(2 )(1)=
17. (b) xo )(t) =
e2'
(1
te2(
+
1
1
( 0
1
(d) x (3) (t) =
e2(
0
—I
(1 2 /2)0 +
—1
0
t + 1
1
(e) kli(t)=e2' (1
—1
—t
')
1
0
1
(f) T= ( 1
2
1
0
0
2
(1
0
18. (a) x111 (t) =
t et
0
0
+
_.1
0
2
—3
—2
—2
1
1
4
2
—2
0
J = (0
1
0
0
er
ou e'
1
T -1 =
0,
- 1
1
2
0
4
2
—2
—1/2
0
1/4
1_
—3/2
2t
4t
—2t — 1
0
0.
—1
1
2x
= (2
0
exp(Jt) =
et
0
0
0
0
2
—3
2:
4t
—2: — 1
2
(1
19. (a) J 3
—1
—1
( 1
0
2
e:
1
2
T1 =
x(21(t)=
e',
4
(d) x(3) (t)=
(f) T=
3
0
J= 0
(e) kl“n =
—(12/2)+3
—3
3
1
)
(12/2)+r
0,
—1
e2(
1 te2' + (23)
0
t +2
;k2)
;. 33i.2
J3 =
'
0
(;.4
.1 =
;0
4;0)
A4
0
1
e'•'
0
1
1
01
)
x = exp(Jt)x°
20. (c) exp(Jt) =
0
t
e:" (0
0
1
et + c2 (3 ) e' +
; (I )
0
21. (c) exp(J0 =
e' ' (
1
t2/2
0
1
t
0
0
1
Secao 7.9
x = c i ( 1I )
+c,
x = c 1 ( 13 e'`
(
) e -2r — (
- — n/-
x = ci
2c os t + sen t
x = c l (2) e -3'
+ C2
teg —
4 ( 31 )e'
2 /3 )
, (— 5sent
COS I + 2sen t
(1) e2I — CO e-2( +
( 21 )
t—
(
, —1
11 Nij C +
4
+
+
(?)
_i
2/0)e
2
1 tcost —
1 ) et
(0
(1)
tsent — 1 cost
1
586
x = ( 21 ) +
C2 [ ( 21 )
- ( )] - 2 ( 21 ) In t + ( 25 ) t -1 -
I
1-2
4 (-20
1 ) e _ 5, + ( 21) in t 38 ( 21) t s
x =+ c (22
1
1
7. x = c 1 ( ) e 3( + c2 ( -2 ) e +
8. x = c
14 ( )
1 \ ± 2 ( 1) ter
() e t + c2 ( 1 ) e-t + ( o)
1
\ _,
9. x=c 1 ( 1 )e -r2 +c2U
-1 e (± (3)t -
I;61
(
_t
15
7 ) + \1/
2
1 VI - I
, 1 ( 2+1 ,
-
1
e
te- +
x = c i ( 7,) e' + c2 (e -4t 1
j. 2 - A
9 -1 - -./2
v L
/
5 cost
x = C1 ()cost +sent + c' -
5sen t
(0
(5/2)
t cost -
tsent - ( 5/2) cos t
t ±2
scot
1
. ) + 1/2)
1
COS
12. x = [-I In(sent) - In(- cost) - 3t2 + 6.11 (
5 cost
/ cost +sent)
5 sent
+ G-• In(sent)- lt + c2] ( -cost 4- 2scnt
13. (a) 11 (1) =
C l/2 cos r
2
4e- Y2 sen 2it
sen I t )e-`/z
-4e -02 cos It
(b) x = C-f/2
sen t
4 4 cos It
I 4 ,
1
I
t
x c () t + c2 ( 1 ) t -I - ( 32 ) + 2 ( 31 ) t - (i)tInt
1
3
\3)
x c, (2 ) t 2 + C2 ( 1 ) t-i
+ (2) +
\2
CAPiTUL 0 8
( -2 ) t 4 - ( 2)
10 \ I)
/
Seca() 8.1
1 (a) 1,1975; 1.38549; 1,56491: 1,73658
1,19631; 1,38335; 1,56200: 1,73308
1,19297; 1,37730; 1,55378; 1,72316
(d) 1.19405; 1,37925; 1,55644; 1,72638
2. (a) 1,59980; 1.29288; 1,07242; 0,930175
1,61124; 1,31361; 1,10012; 0,962552
1,64337; 1,37164; 1,17763; 1,05334
(d) 1,63301; 1,35295; 1,15267; 1,02407
3. (a) 1,2025; 1,41603; 1,64289; 1,88590
1.20388; 1,41936; 1,64896; 1,89572
1,20864; 1,43104; 1,67042; 1.93076
(d) 1,20693; 1,42683; 1,66265; 1,91802
4. (a) 1,10244; 1,21426; 1,33484; 1,46399
1,10365: 1.21656; 1,33817; 1,46832
1,10720; 1,22333; 1,34797; 1,48110
(d) 1,10603; 1,22110; 1,34473; 1,47688
5. (a) 0,509239; 0,522187; 0,539023; 0,559936
0,509701; 0,523155; 0.540550; 0,562089
0,511127; 0,526155; 0.545306; 0,568822
(d) 0,510645; 0,525138: 0,543690; 0,566529
6. (a) -0,920498; -0,857538; -0,808030; -0,770038
-0,922575: -0,860923; -0,812300; -0,774965
-0,928059; -0,870054; -0,824021; -0,788686
(d) -0,926341; -0,867163; -0,820279; -0,784275
7. (a) 2,90330; 7,53999; 19,4292; 50,5614
(b) 2,93506; 7,70957; 20,1081; 52,9779
3,03951; 8,28137; 22,4562; 61,5496
3,00306; 8,07933; 21,6163; 58,4462
8. (a) 0,891830; 1.25225; 2,37818; 4,07257
0,908902; 1,26872; 2.39336; 4,08799
0.958565; 1,31786; 2.43924; 4,13474
(d) 0,942261; 1.30153; 2,42389; 4,11908
9. (a) 3,95713; 5,09853: 6.41548; 7.90174
3,95965: 5,10371; 6,42343; 7,91255
3,96727; 5,11932; 6,44737; 7,94512
(d) 3,96473; 5,11411; 6,43937: 7,93424
10. (a) 1,60729; 2,46830; 3.72167; 5,45963
1.60996: 2.47460: 333356; 5,47774
1,61792; 2,49356: 3,76940: 5,53223
(d) 1,61528; 2,48723: 3,75742: 5,51404
11. (a) -1,45865; -0,217545: 1,05715; 1,41487
-1,45322; -0,180813; 1.05903: 1.41244
-1,43600; -0,0681657; 1,06489: 1.40575
(d) -1,44190; -0,105737; 1,06290; 1.40789
12. (a) 0,587987; 0.791589; 1.14743: 1,70973
0.589440; 0.795758; 1,15693; 1,72955
0,593901: 0.808716; 1,18687; 1,79291
(d) 0,592396; 0,804319; 1,17664; 1.77111
1,595; 2.4636
en+1 = [2fpan)-11h 2 , I en+Il [1 + 2 max0 9 .5 i 10( 1 )1] ,
e„+1 = e2,„112. lei I < 0,012, le4 1 < 0,022
e„,.1=[20(t„)-7„]h2, I e„. 1 1 [1 + 2 max0 ,.: i 10 (0 I] h2,
e„+, = 2e21„,.n2, l e i I < 0,024, 1e4 1 < 0,045
e„+1 = [7„ +
(i „) + q53 (inflh 219. e„,_ 1 =119 - 15/4-1/2(7„)02/4
en , ' = (1 + + 0(701 1121/12/4
e„+1 = -1'(7„) + 2i;;1 expl-7„0(i„)] -7„ expl-27„0(i„)I1//2/2
22. (a) CO= 1 + (1/57r)sen 571
(b) 1.2; 1,0; 1.2
(c) 1,1; 1,1: 1.0: 1,0
,F5()7 0.08
(d) h < 1/.:
e„+1 = -0"(i„)h2
(a) 1,55; 2,34; 3.46; 5.07
1,20; 1,39; 1,57; 1,74
1,20: 1,42; 1,65; 1,90
26. (a) 0
(b) 60
(c) -92,16
27. 0,224 0 0.225
Seciio 8.2
1. (a) 1,19512; 1,38120; 1,55909; 1,72956
1,19515: 1,38125; 1,55916: 1.72965
1,19516; 1,38126; 1,55918; 1,72967
2. (a) 1.62283: 1,33460: 1,12820; 0,995445
1,62243: 1,33386; 1,12718; 0.994215
1.62234; 1,33368; 1,12693; 0,993921
3. (a) 1,20526; 1,42273; 1,65511: 1.90570
1,20533; 1,42290; 1,65542; 1,90621
1,20534: 1,42294; 1,65550: 1,90634
4. (a) 1,10483; 1,21882; 1,34146: 1,47263
1,10484; 1,21884; 1,34147: 1,47262
1,10484; 1,21884; 1.34147; 1,47262
5. (a) 0,510164; 0,524126: 0,542083: 0,564251
0,510168; 0,524135: 0,542100; 0,564277
0,510169; 0,524137; 0,542104; 0,564284
6. (a) -0,924650; -0,864338: -0,816642: -0,780008
-0,924550; -0,864177; -0,816442; -0,779781
-0,924525: -0,864138; -0,816393; -0,779725
7. (a) 2,96719; 7,88313; 20,8114; 55,5106
(b) 2,96800; 7,88755; 20,8294; 55,5758
587
588
RESPOSTAS DOS RPOISLEMAS
(a) 0,926139; 1.28558; 2.40898: 4,10386
(h) 0,925815; 1.28525; 2.40869; 4,10359
(a) 3,96217; 5,10887: 6.43134; 7.92332
(b) 3,96218: 5,10889: 6.43138: 7,92337
10. (a) 1,61263: 2,48097; 3.74556: 5,49595
(b) 1,61263; 2.48092; 3.74550; 5.49589
(a) -1.44768: -0,144478: 1,06004: 1,40960
(b) -1,44765; -0,143690: 1,06072: 1,40999
(a) 0.590897: 0.799950: 1.16653: 1.74969
(b) 0,590906: 0.799988: 1.16663: 1.74992
en+ i = (38h 3 /3) exp(47„), le„ 4. 1 1 < 37. 758 8h3 em 0 < t < 2, l e r I < 0,00193389
e„ +1 = (2h 3 /3) exp(27„), le n+1 1 < 4,92604h3 cm 0 < t < 1, 'e l l < 0,000814269
len+11 < 9,85207h 3 e m 0 < t 1, le i l < 0.00162854
e„+ 1 = ( 4h3 /3)exp(27„).
h--' 0,071
19. /r 0,023
20. h 0,081
21. h 1. =-'• 0,117
23. 1,19512. 1,38120. 1.55909, 1,72956
24. 1,62268, 1,33435, 1,12789, 0.995130
25. 1,20526, 1,42273. 1.65511. 1,90570
26. 1,10485, 1,21886, 1.34149, 1.47264
Seca() 8.3
(a) 1,19516; 1,38127: 1.55918; 1.72968
(b) 1,19516: 1.38127: 1.55918: 1.72968
(a) 1,62231; 1.33362: 1.12686: 0.993839
(b) 1,62230; 1.33362: 1.12685: 0.993826
(a) 1,20535; 1,42295: 1.65553: 1.90638
(b) 120535: 1.42296: 1.65553; 1.90638
(a) 1,10484; 1.21884: 1.34147: 1.47262
(b) 1,10484; 121884: 1.34147: 1.47262
(a) 0.510170: 0.524138: 0.542105: 0 564286
(h) 0.520169: 0.524138; 0.542105; 0 564286
(a) -0,924517: -0,864125: -0 816377; -0.779706
(b) -0,924517: -0,864125; -0 816377; -0.779706
(a) 2,96825; 7 88889: 20.8349: 55.5957
(b) 2,96828; 7 88904; 20.8355; 55,5980
(a) 0,925725; 1.28516; 2.40860; 4.10350
(b) 0,925711: 1.28515; 2.40860: 4.10350
(a) 3,96219; 5,10890; 6.43139; 7.92338
(b) 3,96219; 5,10890: 6.43139; 7,92338
(a) 1,61262: 2,48091: 3,74548; 5.49587
(b) 1,61262: 2,48091: 3,74548: 5.49587
(a) -1,44764; -0.143543: 1,06089: 1,41008
(b) -1,44764; -0,143427: 1,06095: 1,41011
(a) 0,590909; 0.800000: 1.166667: 1,75000
(h) 0,590909; 0,800000; 1.166667: 1,75000
Seca° 8.4
1. (a) 1,7296801; 1.8934697
1,7296802; 1.8934698
1,7296805; 1.8934711
2. (a) 0,993852; 0,925764
0,993846: 0.925746
0,993869; 0,925837
3. (a) 1,906382: 2.179567
1,906391; 2.179582
1,906395; 2.179611
4. (a) 1,4726173; 1.6126215
1,4726189; 1,6126231
1,4726199; 1.6126256
589
5. (a) 0,56428577: 0.59090918
0,56428581: 0.59090923
0,56428588: 0.59090952
6. (a) —0,779693: —0.753135
—0,779692: —0,753137
—0,779680: —0,753089
7. (a) 2,96828; 7,88907: 20.8356; 55,5984
2.96829; 7,88909: 20.8357: 55.5986
2,96831; 7.88926; 20.8364; 55.6015
8. (a) 0,9257133: 1.285148: 2.408595: 4,103495
0,9257124: 1.285148: 2.408595: 4,103495
0,9257248: 1.285158: 2.408594: 4,103493
9. (a) 3,962186; 5.108903: 6.431390: 7.923385
3,962186; 5.108903; 6.431390: 7.923385
3,962186; 5,108903: 6.431390: 7,923385
10. (a) 1,612622: 2,480909: 3,745479: 5,495872
1,612622; 2.480909; 3.745479: 5.495873
1,612623: 2.480905; 3.745473: 5,495869
11. (a) —1,447639: —0.1436281: 1.060946: 1,410122
—1,447638: —0.1436762: 1,060913: 1,410103
—1,447621: —0.1447219: 1.060717: 1,410027
12. (a) 0,5909091: 0.8000000: 1.166667: 1,750000
0,5909091: 0.8000000: 1.166667: 1,750000
0,5909092; 0.8000002: 1,166667: 1,750001
Sec5() 8.5
(b) 02 (0 — 01 (0 = 0.001e:oc quando t
oc
(b) (t) = ln[et /(2 — e)1: 02 (0=111[1/(1 — t)I
(a,b) It = 0,00025 e suficiente. (c) It = 0,005 6 suficiente.
(a) y = 4e-"'s + (r214). (c) 0 maodo de Runge-Kutta 6 estavel pant
(d) It = 5/13 0.384615 t.".! suficientemente pequeno.
6. (a) y = 12
5. (a ) Y = t
It =
0,25. mas 6 instavel Kara h = 0,3.
Seciio 8.6
1. (a) 1,26, 0,76; 1.7714, 1.4824; 2.58991, 2,3703; 3,82374, 3,60413;
5,64246, 5,38885
1.32493, 0.758933; 1.93679, 1.57919; 2,93414, 2,66099; 4,48318, 4.22639;
6,84236, 6,56452
1,32489, 0359516; 1.9369, 1.57999; 2,93459, 2,66201; 4,48422, 4.22784;
6,8444, 6,56684
2. (a) 1,451, 1,232; 2,16 133, 1,65988; 3,29292, 2,55559; 5,16361, 4,7916:
8,54951, 12,0464
1,51844, 1,28089; 2.37684, 1.87711; 3,85039, 3.44859; 6,6956, 9.50309;
15,0987. 64,074
131855. 1,2809: 2.3773, 1,87729; 3,85247, 3,45126; 6,71282, 9,56846;
15,6384, 70,3792
3. (a) 0,582, 1,18; 0.117969. 1,27344: —0.336912, 1.27382; —0,730007, 1.18572;
—1,02134, 1,02371
(b) 0,568451, 1,15775; 0.109776. 1,22556; —0,32208, 1,20347;
0,681296, 1.10162; —0,937852. 0.937852
(c) 0,56845, 1,15775; 0,109773, 1.22557; —0,322081, 1,20347;
—0,681291, 1,10161; —0.937841, 0.93784
4. (a) —0,198, 0,618; —0,378796, 0.28329; —0.51932, —0,0321025;
—0,594324, —0,326801; —0,588278, —0,57545
—0,196904, 0,630936: —0,372643, 0,298888; —0,501302, —0,0111429;
—0,561270, —0,288943; —0,547053, —0,508303
—0,196935, 0,630939; —0,372687, 0,298866; —0,501345, —0,0112184;
0,561292, —0,28907; —0,547031. —0,508427
590
(a) 2,96225, 1,34538; 2,34119, 1,6712 1; 1,90236, 1,97158; 1,56602, 2,23895;
1,29768, 2,46732
(b) 3,06339, 1,34858; 2,44497, 1,6863 8; 1,9911, 2,00036; 1,63818, 2,27981:
1,3555, 2,5175
(el 3.06314, 1,34899; 2,44465, 1.6869 9; 1,99075, 2,00107; 1,63781, 2,28057;
1,35514, 2,51827
(a) 1,42386, 2,18957; 1,82234, 2,3679 1; 2,21728, 2,53329; 2,61118, 2,68763;
2,9955, 2,83354
1,41513, 2,18699; 1,81208, 2,3623 3; 2,20635, 2,5258; 2,59826, 2.6794;
2,97806, 2,82487
1,41513, 2,18699; 1,81209, 2,3623 3; 2,20635, 2,52581; 2,59826. 2.67941;
2,97806, 2,82488
Para h = 0.05 e 0,025: x = 10,227,y = -4,9294; estes resultados esttio de acordo corn a
soluciio exata ate cinco digitos
9. 1,99521, -0,662442
1,543, 0,0707503; 1,14743, -1,3885
CAPITULO 9
Seca° 9.1
= (1, 2) T; r2 = 2, V) = (2, 1) T (b) ponto de sela, instavel
(a) r, = -1,
2,
r
=
(1,
3) T; r2 = 4. V) = (1, 1) r(b) no, instavel
(a) r, =
(a) r, = -1, r = (1, 3) T; r2 = 1, V" = (1, 1) r (b) ponto de seta, instavel
(a) r, = r2 = - 3; V) = (1, 1) T (b) no imprOprio, assintoticamente estavel
= (2 f i. 1) r(b) ponto espiral, assintoticamente estavel
(a) r„ r, = -1 ± i;
44 °,44.2) = ( 2 ± i, 1) r (b) centro, estavel
(a) r,, r, =
(b) ponto espiral, instavel
(a) r,, r, = 1 ± 2i; r, 412) = (1, 1
(a) r, = - 1,4( 1 ) = (1, 0) T; r, = -1/4, V) = (4. -3) r (b) nO, assintoticamente estavel
(a) r, = r, = 1;1`" = ( 2. 1) r (b) no imprOprio, instavel
= (2, -1 ± 30' (b) centro, estavel
(a) r,, r, = ± 3i;
(a) r, = r2 = -1;4" = ( 1, 0) r, V) = (0, 1) T (b) no prOprio, assintoticamente estavel
(a) r r2 = ( 1 f 31)12;1;"", 4" 2) = (5, 3 3i) r(b) ponto espiral, instavel
= -4; ponto do sela, instavel
xo = 1, yo= 1; r, =
xo = -1, y„= 0; r, = -1, r2 = - 3; nO, assintoticamente estavel
ponto espiral, assintoticamente estavel
±
x„ = -2, y„= r„ r2 =
xo = y/S,yo = al r,, r, = IFSi; centro, estavel
17. c2 > 4km. nO, assintoticamente estavel; = 4km . no imprOprio, assintoticamente estavel; c= < 4km, ponto espiral, assintoticamente estavel
Seca° 9.2
x =
, y = 2e-2`, y x2/8
x = 4e-`, y = 2e2t , y = 32x -2 ; x = 4e-`, y = 0
16
x = 4 cos t, y = 4 sen t, .v2 + y2 = 16; x = -4 sen t, y = 4 cos t, x2 +
x = fa cos ,i7d)t, y = - f sen r7)t; ( x2 /a) + (y2 / b) = 1
(a, c) (4, 1), ponto de sela, instavel; (0, 0), no (prOprio), instavel
(a. c) (-0/3, -+), ponto de sela, instavel; (0/3, - centro, estavel
(a, c) (0, 0), n6, instavel; (2, 0), nO, assintoticamente estavel; (0, 4), ponto de sela, instavel; (-1, 3). nO,
assintoticamente estavel
(a, c) (0, 0), nO, assintoticamente estavel; (1, -1), ponto de seta, instavel; ( I, -2), ponto aspiral, assintoticamente estavel
(a, c) (0, 0), ponto espiral, assintoticamente estavel; (1 -4, 1 +4), ponto de sela. instavel; (1 +4, 1
-4), ponto de sela, instavel
(a, c) (0,0), ponto de sela, instavel; (2, 2). ponto espiral, assintoticamente estavel; (-1, -1). ponto espiral,
assintoticamente estavel; (-2, 0), ponto de sela, instavel
(a, c) (0, 0), ponto de sela, instavel; (0. 1). ponto de sela, instavel; (+, ÷), centro, estavel; (-4, 4), centro,
estavel
0), ponto
(a, c) (0, 0), ponto de sela, instavel; (A. 0), ponto espiral, assintoticamente estavel; espiral, assintoticamente estavel
(a, c) (0, 0), ponto de sela, instavel; (-2. 2), no, instavel; (4, 4), ponto espiral, assintoticamente estavel
(a, c) (0, 0), ponto de sela, instavel; (2, 0), ponto de sela, instavel; (1, 1), ponto espiral, assintoticamente
estavel; (-2, -2), ponto espiral, assintoticamente estavel
15. (a, c) (0, 0), nO, instavel; (1, 1), ponto de sela, instavel; (3, -1), ponto espiral, assintoticamente estavel
y2 =
RLSPOSTAS DOS PROBLEMAS
591
(a, c) (0, 1), ponto de sela, instavel; (1,1), no, assintoticamente estavel; (-2, 4), ponto espiral, instavel
18. (a) 4x2 + y2 = c
(a) 4x2 - y2 = c
20.
(a) arctan(y/x) - In
y2 = c
19. (a) (y - 2x)2 (x + y) = c
=
c
22.
(a)
x2y2 - 3x2y - 2y2 c
21. (a) 2x2y - 2xy + y2
2
4
=c
23. (a) (y2 /2) - cos x = c
24. (a) x
+ y2
(x
/12)
Seciin 9.3
Linear e no linear: ponto de sela, instavel
Linear e tido linear: ponto espiral, assintoticamente estavel
Linear: centro, estavel; nâo linear: ponto espiral ou centro, indeterminado
Linear: no imprOprio, instavel; nä° linear: no ou ponto espiral, instavel
ponto de sela, instavel
(a, b, c) (0, 0); ti' = -211 + 2v, v' = 4u + 4u; r = I ±
(-2, 2); it' = 4u, v' = 6u + 6v; r = 4, 6; n6, instavel
ponto espiral, assintoticamente estavel
(4, 4); it' = -6ii + 6u, v' = -8u; r = -3 ±
(a, b, c) (0, 0); u' = u, v' = 3v; r = 1,3; nO, instavel
(1, 0); u' = -u - v, v' = 2v; r = -1, 2; ponto de sela, instavel
v' = (- )u- 3v; r =
-3; nO, assintoticamente estavel
(0, 4); u' =
- 4v; r = (-3 ± /f7)/2; ponto de sela, instavel
+ r, v' =
(-1, 2); u'
(a, b, c) (1, 1); u' = -v, v' = 2u - 2v; r = -1 ± i; ponto espiral, assintoticamente estavel
ponto de sela, instavel
(-1,1); u' = -v. c' = -2u - 2v; r = -1 ±
(a, b, c) (0, 0); u' = u, v' = ( ,)v; r = 1. 4-; nO. instavel
no, assintoticamente estavel
= (-4)u - (4) • ; r = -1,
(0, 2); fi' =
(I, 0); u' = -u - r. v' = (-4)c; r = -1. -1/4; nO, assintoticamente estavel
(+, ); u' = (-)u - (+)v, v' = (q)u -
r = (-5 ± ../57)/16; pont° de sela, instävel
ponto de sela. instavel
+ 2v, v' = u + 2v; r = (1 ±
9. (a, b. c) (0, 0); is' =
+ 3u, = -2u; r = (-3 ± j377i)/4; ponto espiral, assintoticamente estavel
(2, 1); is' =
centro ou ponto espiral, indeterminado
(2, -2); ii' = -3v, v' = it; r =
(4, -2); =
= -u - 2v; r = -1 ± ,f5; ponto de sela, instavel
10. (a, b, c) (0, 0); is' = is, v' = v; r = 1. nO ou ponto espiral, instavel
(-1,0): u' =
= 2v; r = -1, 2; ponto de sela, instavel
(a, b, c)(0, 0); u' = 211 + v, v' = u -
r = ±./3: ponto de sela, instavel (-1,1935; -1.4797): is' = -1,2399u
- 6,8393v, v' = 2,4797u - 0,80655v; r = -1,0232 ± 4,1125i; ponto espiral, assintoticamente estavel
(a, b, c) (0, ±2n r), n = 0, 1, 2, ...; it' = v, = -is; r = ±i; centro ou ponto espiral, indeterminado
(2, ± 2(n - 1)7), n = 1, 2, 3, ...; is' = -3v, v' = -ii; r = ±173; ponto de seta. instavel
(a, b, c) (0, 0); is' = if, v' v; r = 1, 1; n6 ou ponto espiral, instavel
( I, 1); u' = is - 2v. v' = -2u + v; r = -1; ponto de sela. instavel
(a, b, c) (1, 1); is' = - v, v' = u - 31.; r = -2. -2; no ou ponto espiral, assintoticamente estavel
(-1,-1); = ii + u, v' = is - 3v; r = -1 ± .13; ponto de sela, instavel
- v, v' = is - u; r = (-3 ± 00/2; ponto espiral, assintoticamente estavel
(a, b, c) (0, 0); =
(-0,33076; 1,0924) e (0,33076; -1.0924); = -3,5216u - 0,27735v, v' = 0,27735u + 2.6895v; r = -3,5092;
2,6771; ponto de seta, instavel
(a, h, c) (0, 0); is' = u + v, v' = -is + v; r = 1 ± is ponto espiral, instavel
+ (;,)i.f; r = (7 ± ./273)/4; ponto de sela, instavel
17. (a, b, c) (2, 2); is' = -4u, v' =
(-2, -2); is' = 4v, = (4)" - )v; r = (-1 ± 33)/4; ponto de sela, instal/el
= -4v, = (2 )u; r = ±../171i; centro ou ponto espiral, indeterminado
(-*, -2); is' = 4v. v' = (-2)it; r = ±4i; centro ou ponto espiral, indeterminado
18. (a, b, c) (0, 0); is' = 2fi - v, v' = 211- 4v; r = -1 ± Nr7; ponto de sela, instavel
(2,1); u' =
v' = 411 - 8v; r = -2, -6; no, assintoticamente estavel
(-2, 1); u' = 5u, v' = -4u; r = ±2,./3i; centro ou ponto espiral, indeterminado
(-2, -4); it' =10if - 5v, v' = 6ii; r = 5 ±
ponto espiral, instavel
21. (b, c) Veja a Tabela 9.3.1
(a) R = A, T 3,17
(b) R = A, T 3,20; 3,35; 3,63; 4,17
quando A -+ 0
(c) T
(d) A =
(b) tic 4,00
4,51
25. (b)
dx/dt
= y, dy/di = -g(x) - c(x)y
30. (a)
0 sistema linear é dx/dt = y, dy/dt = - g' (0)x - c(0)y
Os autovalores satisfazem r2 + c(0)r + g'(0) = 0
3
.
(-2-,2),
77"
592
Secäo 9.4
(b, c) (0, 0); u' = (i37 )u, v' = 2 • ; r = -7;. 2: n6. instavel
- 2v; r = -2: ponto de sela. instavel
(0, 2); u' = (4)0, v' =
i; ponto de sela, instavel
v' = (Dv: r =
(3, 0); u' = (4)u -
(1, 25-); it' = (-4-)u - ( 0, v' = (-)u - (1)v; r = (-22 ± ,,,75:74)/20; no, assintoticamente estavel
(b, c) (0, 0); u' = (4)u, v' = 2v: r = 4, 2: nO. instavel
-2: nO, assintoticamente estavel
=
- 2r: r =
(0, 4); u' =
4; nO, assintoticamente estavel
(1,0); u' =
- (4)v, v' = (-+)v; r =
v' = (-1,)u - (+)v; r = (-3 ± 10)/4: ponto de sela, instavel
(1, 1); u' = -u -
3. (b, c) (0, 0); it' = (3)u, v' = 2v; r = 4.2: nO. instavel
-2; nO. assintoticamente estavel
(0, 2); u' = (4)u, V = (-1)u - 2r: r =
-7 : nO. assintoticamente estavel
1 : r =
0); u' = (-Du - 3v, = (- 4)1:
71 )u - (1-1-10v; r = -1.80475:0,30475; ponto de sela, instavel
+61-); u' = (-1)u - (1)
v , v' = (- 87
4. (b, c) (0, 0); u' = (12 )u, v' = ( !4 )v; r = 2: no, instavel
(0, 4); It' = (4)u, = (4)v; r = ±4, ponto de sela, instavel
(3, 0); u' = (-4)u 3v, = (4)v; r = -4. ponto de sela. instavel
(2, -;); It' = x - 2v, v' = (4). )11 - ()t . ; r = -1.18301; -0.31699; nO, assintoticamente estavel
nO. instavel
(b, c) (0, 0); u' = u, = (4)v; r =
nO, assintoticamente estavel
1
=
(2.
,)tt
(1,)v:
r
=
=
(0, 2); tt'
2
r = -1. J.7 : ponto de sela. instavel
- v, =
(1, 0); it'
(b, c) (0, 0); ri' = u, = (4)1 . ; r 1. 4, : n6, instavel
-512; ponto de sela, instavel
z'' = (M u - (4)1 . : r =
( 0 ,3): u ' =
(-.)v, =
r = - 1 . 4 : ponto de sela. instavel
0); tr' =
r = (-5 -1- 0)/2; n6, assintoticamente estavel
2); u' = -2u + v, = (4)11 -
0,y
(a) Os pontos criticos sac) x = 0,y = 0:x = ,la,, y = 0;x = 0,y = E
6,1a, quando t
oo; os
vermelhOes sobrevivem
c,la,, y
0 quando
(h) Os mesmos pontos criticos que em (a), mas
os peixes azulaclos
sobrevivem
(a) X = (B - y,R)I(1 - y,y). =(R - K.B)I( I - y,y)
(h) X diminui, Y aumen la; sum. se B se tornar menor do que y,R , en tao x
e y R quando t oo
10. (a) al c2 - a2 e 0: (0, 0), (0, E . ,/a2 ). ( 1 /(71 . 0)
a2E, = 0: (0, 0) e todos os pontos na reta a ,x + a, y (t
o-,E 2 - a,( 1 > 0: (0, 0) é urn 116 instavel: (6,/a1 .0) 6 urn ponto de sela: (0. 6 21(72 ) e um nO assintoticamente estavel
a ,c, - a2c, < 0: (0,0)6 urn nO instavel: (O. E/a2 ) 6 urn ponto de seta: (e,/a,, 0) 6 um nO assintoticamente estavel
(0,0) 6 urn nO instavel; os pontos na reta a,.r + a ,y = c, silo pontos criticos estaveis, nal° isolados
(a) (0, 0), ponto de sela; (0,15; 0). ponto espiral se y 2 < 1,11; (2, 0), ponto de sela
(c) y ..==, 1,20
(b) (2- '4 - a , ;a), (2 + \/4 - a, a)
(1, 3) 6 um nO assintoticamente estavel; (3,3) 6 um ponto de sela
a„ = 8/3; o ponto critico 6. (2, 4): = 0. -1
14. (b) (2 - J4 - ;a, ice), (2 + \/4 - a, ia)
(1, 3) 6 urn ponto de sela; (3,3) e urn ponto espiral instavel
a„ = 8/3; 0 ponto critico e (2. 4); ?. = 0.1
15. (b) ([3 - N/9 - 4a]/2, [3 + 2a -
- 4a]/2),
([3 + .39 - 4a]/2, [3 + 2a +
- 4a)/2)
(1, 3) 6 urn ponto de sela; (2, 4) é urn ponto espiral instavel
a„ = 9/4;0 ponto critico é (3/2,15/4): = 0, 0
16. (b) ([3 - - 4a]/2, [3 + 2a - - 4a1/2),
([3 +
- 4a1/2, [3 + 2a + ./9 - 4a]/2)
(1, 3) é urn centro da aproximaciio linear e tambem do sistema linear; (2, 4) e urn ponto de sela
a„ = 9/4; 0 ponto critico c (3/2. 15/4): = 0, 0
17. (b) P,(0, 0), PALO), P,(0, a). P,(2 - 2a, -1 + 2a). P, esta no primeiro quadrante para 0.5 < a < 1.
a = 0; P3 coincide corn P,. a = 0,5; P, coincide corn P2. a = 1; P4 coincide corn P3.
1 - 2x - y
-x
= ( -0,5y
a - 2y - 0.5x
593
(e) P, é urn no instavel para a > 0. P, 6 urn no assintoticamente estavel para 0 < a < 0,5 e urn ponto de
sela para a > 0,5. P3 6 urn ponto de sela para 0 < a < 1 c um no assintoticamente estavel para a > 1. P,
urn no assintoticamente estavel para 0.5 < a < 1.
18. (h) P,(0, 0), P,(1, 0). P,(0; 0,75/a). P,R4a - 3)/(4a - 2), 1/(4a - 2)]. P, esta no primeiro quadrants para
. 0,75.
(c) a = 0,75; P3 coincide corn P.
- 2x - y
-x
(d) =
-0.5y
0.75 - 2av - 0,5x )
(e) P, 6 urn no instavel. P. é um ponto de sela. P, 6 urn no assintoticamente estavel para 0 < a < 0,75 e
urn porno de sela para a > 0,75. P, e urn no assintoticamente estavel para a > 0,75.
19. (b) P,(0, 0), P2 (1,0), P3(0, a), P.,(0.5: 0,5). Alem clisso.para a = 1, todo ponto na reta x+y=16 urn ponto
critico.
a = P3 coincide corn P,.Tambem a = 1.
J =
2.v - y
-(2a - 1)y
( 1 -
-x
a - 2y - (2a - 1)x
(e) P, é urn no instavel para a > 0. P,e P3 sào pontos de sela para 0 < a < 1 e nos assintoticamente estaveis para a > 1. P, 6 um ponto espiral assintoticamente estavel para 0 < a < 0,5, urn n6 assintoticamente
estavel para 0.5 < a < 1 e urn porno de sela para a > 1.
Seciio 9.5
(b. c) (0. 0); u' = (4)u, = (--1.7 )v; r = 4. - ponto de sela. instavel
(1, 3); u' = ( - +)v, e' = 3u; r = ±NA 2; centro ou ponto espiral, indeterminado
(b, c) (0, 0); u' = It, = ( - 1- )v; r = 1. - ponto do sela. instavel
( 7 , u' -
t = it, r = ±( 7 )1.s.cntro ou ponto espiral, indeterminado
(b, c) (0, 0); u' = u, = (-4)v; r = 1. - ; pont° de sela. instavel
ponto de sela. instavel
(2, 0); u' = - it - v, v' = (4)1 .; r =
. 10/8; ponto espiral, assintoticamente estavel
V = (4)1t: r = ( - 1VT
-
(1, 4); u' =
(b, c) (0, 0); u' = ()1 t. = - v; r .4. -1: ponto de sela. insuivel
porno de sela, instavel
0); u' = ( 4 )11 - ()v. = ()t.; r =
(1, +); u' = - u - (4)r. = ( .IT )u:r = (-1 ± .7075)/2; nO. assintoticamente estavel
(b, c) (0, 0); u' = - u, v' = (-4)v; r = -1.-4; n6. assintoticamente estavel
0); u' = (4)u -
= - v; r = -1. 4; porno de sela, instavel
0); u' = -3u - (4)v. v' = (1)v; r = -3. 4, ; porno de sela, instavel
(4. 4): u' = ( - 4)11 - (;)v, = (4)tt: r = ( - 3 ± 01)0/8; porno espiral, assintoticamente estavel
um maxima
//Pith 6 um maxima
(b. c)
t = 0, T, 2T....:
P 6 um maximo.
dH/cit e urn minimo.
t = T/4, 5T/4. ...:
d PI dt e um minima
= T/2, 3T/2. ...:
H 6 urn minima
P c urn minim°.
t = 3 774, 7 774, ...:
dfildt 6 urn maxim.
(a) .,,/a/„/iiy
(b) ,J
(d) A raziio das amplitudes da presa e do predador aumenta bem devagar quando o ponto inicial se
afasta do ponto de equilibrio.
(a) 42r/0 1' 7.2552
(c) 0 perioclo aumenta devagar quando o ponto i n icial se afasta do ponto de equilibria
(a) T 6,5 (b) T 3,7, 7' 11.5 (c) T 3,8, T 11,1
II. (a) P1 (0, 0). P,(1/a, 0), P3(3,2 - 6a): Pz se move para a esquerda e P, se move para baixo; eles coincidem
em (3, 0) quando a = 1/3.
(b) P, 6 urn ponto de sela. P, e urn porno de sela para < 1/3 e um no assintoticamente estavel para a
> 1/3. P3 é urn ponto espiral assintoticamente estavel para < a, = (,/7/3 - 1)/2 0,2638, um no assintoticamente estavel para a, < a < 1i3 e um ponto de sela para a > 1/3.
(a) P,(0, 0), P,(ala. 0), P,[cly,(ala)- (calay)]: P, se move paraa esquerda e P3 se move para baixo; eles
coincidem em (sly, 0) quando a = aylc.
(h) P, é urn ponto de sela. P2 6 urn ponto de sela para a «tylc e um no assintoticamente estavel para a >
aylc. P, 6 um ponto espiral assintoticamente estavel para valores suficientemente pequenos de a e torna-se
urn no assintoticamente estavel em urn determinado valor a, < nylc. P3 e urn ponto de sela para a > aylc.
(a, h) P1 (0,0) é urn ponto de sela; P,(5, 0) é urn ponto de seta; P3(2; 2,4) 6 urn ponto espiral assintoticamente estavel.
I.
14. (b) A mesma populacao de presas, menos predadores
Mais presas, a mesma quantidade de predadores
Mais presas, menos predadores
15. (b) A mesma populacao de presas, menos predadores
Mais presas, menos predadores ainda
16. (b) A mcsma populacao de presas, menos predadores
Mais presas, a mesma quantidade de predadores
Secan 9.7
2. r = 1.6 = -t + to, ciclo limite semiestavel
1. r = 1,0 = t + to, ciclo limite estavel.
3,0
=
t + to, solucao periOdica instavel
r
=
ciclo
limite
estavel;
=
t
+
to,
1,0
r=
r = 1, 0 = -t + t0, soluc5o periOdica instavel: r = 2, 0 = -t + t0. ciclo limite estavel
r = 2n - 1, 0 =1+ to,n = 1, 2, 3, ..., ciclo limite estavel
r =2n,0= t + to,n = 1, 2,3, ..., solucdo periOdica instavel
r = 2, 0 = -t + to, ciclo limite semiestavel
r =3,0 = -t + to, solucrio periOdica instavel
8. (a) Sentido trigonometric°
(b) r = 1, 0 = t + ciclo limite estavel; r = 2, 0 = t + t„, ciclo limite semiestavel; r = 3, 0 = t + to, solucdo
periOdica semiestavel
9. r = f,e = -t + to, solucrio periOdica instavel
6,29;
= 1, T= 6.66; µ= 5. T= 11.60
(a) = 0,2, T= 6,29;
(a) x' = y, y' = -x kty - gy3/3
0 < p < 2, ponto espiral instavel; p 2, no instavel
A "L-' 2,16, 7' ..--_. 6,65
(d) p = 0,2, A L. 1,99, T L' 6,31; it = 0.5, A L-. 2.03, T L 6,39;
4,36, T L 11,60
= 2, A L. 2,60, T 7,65; p = 5, A
16. (h) x' = px + y, y' = -x + ;Ay; A = p f i; a origem e um porno espiral assintoticamente estavel para p <
0 e urn ponto espiral instavel para p > 0
(c) r' = r(it - r2 ), 0' = - 1
17. (a) A origem é um no assintoticamente estavel para < -2. um porno espiral assintoticamente estavel
para -2 < < 0, urn ponto espiral instavel para 0 < < 2 c urn no instavel para p > 2.
(a, h) (0, 0) é urn porno de seta; (12, 0) é urn ponto de seta; (2, 8) 6 um ponto espiral instavel.
(a) (0, 0), (5a, 0), (2,4a - 1,6)
(b) r = -0,25 + 0,125a ± 0,25./220 - 400a + 25a 2 ; a0 = 2
20. (b) n = [-(5/4 - b) ± ,/(5/4 -
b) 2 -
1] /2
(c) 0 < b < 1/4: no assintoticamente estavel; 1/4 < b < 5/4: ponto espiral assintoticamente estavel; 5/4 <
b < 9/4: ponto espiral instavel; 9/4 < 6: no instavel.
(d) b0 = 5/4
21. (b) k = 0, (1,1994, -0,62426); k = 0,5, (0,80485, -0,13106)
ko 0,3465, (0,95450, -0,31813)
k = 0,4, T L 11,23; k = 0,5, T "L' 10,37; k = 0,6, T L-.. 9,93
(e) k1 L: 1,4035
Seca° 9.8
1. (b) = Ai, jai = (0,0,1) T ; A = A 3 , (2) = (20,9 - 381 + 40r,O)T;
A = A3 , e ) = (20,9 + ,/81 + 40r, 0) T
-2,6667, e l) = (0, 0,1) T ; A2 -22,8277, (2) (20; -25,6554;0)';
(c)
11,8277, (3) L' (20;43,6554;0)T
2. (c) A 1 :4' -13,8546; A.2,
0,0939556 ± 10,1945i
5. (a) dV/dt = - 2a[rx2 + y2 b(z - r) 2 - br21
11. (b) c = 0,5 : P1 (4/4, -4, 4); L = 0,-0,05178 ± 1,5242i
= 0,1612; -0,02882 ± 2,0943i
c = 1 : P1 = ( 0,8536; -3,4142;3,4142);
A = -0,5303;-0,03665 ± 1,1542i
P2(0, 1464 ; -0,5858; 0,5858);
= 0,1893; -0,02191 ± 2,4007i
12. (a) Pi (1,1954; -4,7817; 4,7817); P2(0,1046; -0,4183;0,4183); A. = -0,9614;0.007964 ± 1,0652i
(d) T, L' 5,9
RESPOSTAS DOS PRE:I I:SUMAS
(a,b,c) c 1 "=' 1,243
= 0,2273; -0,009796 ± 3,5812i
(a) P1 (2,9577; -11,8310;11,8310);
P2(0,04226: -0,1690;0,1690); n = -2,9053:0.09877 ± 0.9969i
(c) T2:4: 11,8
15. (a) P1(3,7668: -15,0673:15,0673); A = 0,2324:-0.007814±4.007$i
P2(0.0 3318 : -0,1327; 0,1327); n = -3,7335:0.1083 ± 0,9941i
(b) T4 L- 23,6
CAPITULO 1 0
2
Secao 10.1
2. y = (cot ../27r cos fr + sen
1. y = - sen x
y = 0 para todo L: y = c, sen x se sen L = 0
nao solucâo se cos L = 0
y = - tan L cosx + senx se cos L
6. y =
serhlx + xsen../17r) /2 seni2.7r
5. Nao tern solucäo
8. y = c2 sen2x + senx
7. Nao tern solucao
10. y
cosx
9. y = c1 cos 2x + 3cosx
12. y =
e(1 — e 3 ) x -I Inx +
11. y = - 5 + 3 2
Nä° tern solucao
1, 2, 3, ...
y„(x) = sen[(2n - 1)x/21: n
n.„ = [ ( 2n - 1)/2] 2 ,
A„ = [(2n - 1)/21 2 , y„(x) = cos[(2n - 1)x/2]: n = 1.2,3, ...
no = 0, yo(x) = 1; An = n 2 , y„(v) = cos nx: n = 1,2, 3, ...
An = [(2n - 1)7/2L] 2 , y„(x) = cos[(2n - 1).7x/2Lj; n = 1,2, 3, ...
no = 0, yo(x) = 1 : A,, = (tur L) 2y„(x) = cos(n7rx/L); n = 1, 2, 3, ...
n„ = -[(2n - 1)7r/2/] 2 , y„(x) = sen[(2n - 1)7x/2/..]: n = 1, 2, 3, ...
n„ = 1 + (n7r/ In L) 2y„(x) = x sen(n g In In L): n = 1, 2,3, ...
(c) Q e reduzido a 0,3164 de seu valor original
(a) w(r) = G(R2 - r 2 )/4p.
(a) y = k(x 4 - 2Lx 3 + L3x)/24
y = k(x4 - 2Lx3 + L2x2)/24
y = k(x4 - 4Lx 3 + 6L2x2)/24
Seca() 10.2
2. T= 1
1. 7' = 27r/5
4. T 2L
3. Nao e periklica
6. Nao e periOdica
5. T = 1
8. T = 4
7. T = 2
f (x) 2L - x em L <x < 2L: f (x) = -2L - x em - 3L <x < - 2L
f (x) = - 1 em 1 < x < 2; f (x) = - 8 em 8 < x < 9
11. f = L - x em - L <x < 0
1
2
sen[(2n - 1)7rx/L]
rx
2L( - 1)"
14. (b) f (x) =
13. (b) f (x) = —
sen 2
7r
2n - 1
7 n=1 11
E
[2 cos(2n - 1)x (-1)"Isennxi
7r
15. (b) f (x) = - - + Y.'
+
I?
4 1,,,I
—,
7 r (2n - 1)2
1
4 cc cos(2n - 1)7rx
16. (b) f (x) = - + —
2 72 n=1
(2n - 1)2
E
cc
(b) f(x) = 3L + ., f 2 L cos[ (2n - 1).Trx/L] : (-1riLsen(n:rx/L)1
4
•L'
n_ 1 L
nrr
(2n - 1) 2 7r 2
(b) f (x) =-
E - - .r. cos nitT
[
n=1
+
(2
) 2
rurx
n7r]
sen— sen 2
'
4 ,--,
'". sen[(2n - 1)7x/2]
2
0. 19. (b) (h)
f (x) = - _,
2,1 - 1
7r n=L
2
f (x) rr
E
D.-
n=1
nrr x
8 °° (-1)"
2
cos 2
(b) f (x) r_-_- -3- + 772
n=1
12
(b) f (x) = + -7.
1
v-,
cc
L2n=1
E
cos[(2n - 1)irx/21
2 "'" (-1)"
nrr x
sen 2
+ r-t.
n
(2n - 1)2
,i=i
on+1
sen nnx
595
596
RESFOSTAS DOS PROBLEMAS
[4[1 - (-1)n ] (-1)" -1, im x
1 `.-c- ( — 1) n —
1 1
tur i scn
5 cos ng A' + t
--;
(b) f (x) = —
2
n3;73
2
n2
12 + 7r z L-'
n.--:
I
n=1
108(-1)"
°`" [162[(-1)" - 1] 27(-1)" 1 cos nix
9 x--,
+ 54 sen n rrx
(b) f (x) = i-3 ± 2_,
n2,72
4
4
3
n3 73
n 71r=i
= 1
26. (b) m = 27
= 81
25. (b)
28. f f (t) di pode nä() ser periOdica; pot exemplo. seja f (t) = 1 + cos t
Seca° 10.3
4 e"" sen(2n - 1);rx
(a) f (x) = - E 21z - 1
n=1
(a) f(x) = -7 4
[
°‘‘
2
(2n E
I
L 4L
3. (a) f(x) = - + n2
7 2
2
5. (a) f(x) = - + 2 7r
(-1 r
tz.v
seni
cos[(2n - 1)7rx/L]
On - 1)2
, cos n7 r x
E (-1)"+1
n
4
2
4. (a) f (x) = - +
3 7r-
1
n=1
027r cos(2n - 1)x +
n= 1
E (-1)
2 n - 1
"
I
cos(2n - 1)x
ao
(a) f (x) = -- + E(a„ cos frig x + 1) „ sen 117T X);
2
n=1
2(-1)",
1 -11nn-,
1
On =
,
ao = -..:.- . an = ti-7r2
1/n7r - 4/,1373,
..)
7T
rl
-cost
E
7r
(— 1 )"
sennx
cos frIX
I
(a) f (x) = - - +
n
4 n= I 7r 11 2
n 10; maxlel = 1,6025 cm .v = ±7r
it = 20; maxlel = 1,5867 em .v ±7r
n = 40; maxlel = 1,5788 em x .+7r
Nao 6 possivel
1 - cos tur
2
1
cos n7I X
8. (a) f (x) = 2 +- _,
n2
72 n= 1
(b) n = 10; maxlel = 0,02020 em x = 0,±1
n = 20; maxlel = 0,01012 em x = 0,±1
= 40; maxlel = 0,005065 em x 0,±1
n = 21
c'c (-1)"
2x-,
SCIIMTX
f(1 ) = Z—,
Ir
2
it
a par
impar
n= 1
n = 10, 20, 40;
Não a possivel
maxlel = 1 emx ± 1
tur.v
Mr X
2 cos trr
an)
Sen—
COS
+ [6(1 ncos
2;72
2
n71
2 n=1
n = 10; suplel = 1,0606 quando x ->
= 20; suplel = 1,0304 quandox -> 2
n = 40; suplel = 1,0152 quando X —> 2
Näo 6 possivel
2 - 2 cos nit n- T -2 cos flit
[2 cos nit
senturx
I I. (a) f (x) = + E n271,2 cos n;TX
6 n=1
n = 10; suplel = 0,5193 quandox ->
= 20; suplel = 0,5099 quandox ->
n = 40; suplel = 0,5050 quandox
Nä° is possivel
f (x) = -
n= I
= 0,001345 em.v = ±0,9735
n = 10:
n = 20: maxlei = 0,0003534 cm x ±0,9864
maxlel = 0,00009058 em x = ±0,9931
= 40:
n=4
2
13. y = (co set) nt - nsencot)1 (0)2 _ n2 ), (02n
2 = n2
w
y = (sennt - nt cos n0/2n2 ,
14. y = E b„(w sennt - nsenwt)1 w(0)2 n2.,)
w 0 1.2, 3, ...
11=1
y =
E
b „(tn sen nt - n sen Int) I m (m 2 n 2 ) + h„, (sen nu - nu cos mt)12m 2 ,
(0
n
aftn
4
4
)
'` (- 1"
ny sen turx
12
= - E
12. ( a ) f
1
Y = 7r E
n=
.2 _
r
1
(2n - 1) 2 L 2 - 1
1
sen(2n - 1)t - 1 semi)/
w
c cos(2n - 1)71 - cos cot
4 t
1
y = cos cot + -(1 - cos cot) +
, I (2n - 1)2 [0; - (2n - 1)27r21
II
Secüo 10.4
I. Impar
3. Impar
5. Par
2. Nenhuma das duas
4. Par
6. Nenhuma das duas
1
ti
4
f (x) = 4 + 21. E
1 - cos(n7r/2) cos mr x
11=1
‘---,
'"
f(x) = -,_,
7r- 1
2
n2
mr x
sen(tur /2)
sen n2
2
(nrr 12) -
n=
77
2 '" (-1)"-- 1(2tz - 1)7rx
1
cos (a) f(x ) = 2 + E 2
2n - 1
n.1
22
(a) f (x) = >2
n.1
nn ( -
cos
n:r x
nn )
mr + - sen- sen
lig
2
18. (a) f(x) =
(a) f(x) = 1
19. (a) f (x)
2 (
>2
n=1
20. (a) f (x) _
4
rr
n= 1
2n7r
nx
cos - + cos -- 2 cos n7r sen
3
3
3
P VT
1
I
senburx
2
7r
n
y
, 1
7r
1,-c ,° 2 7r
(a) f(x) = - + 7r
- 2_ -
'Ix
sen -
n=1
2
n 2 fir
nx
4
+ - (cos - - 1) cos 2
2
(a) f(x) = 2E ( - 1 ) n sen fix
..1
(a) f(x) = E IL
n=i
4n2 7r 2 (1 + COS mr) 16(1 -cos n:r)1
nrr x
+ se n
ns7r3n3 rri
2
4
16 N--,
'"' 1 + 3 cos nn
(a) f (x) = 3 + 77. .-,
n2
tur x
cos -
4
n=1
- 1 _ cos rur
6E
3
(b) g(x) = - + 772
n2
'v=1
tin x
6 %--, 1
sen h(x) =
3
2
L -n
n=1
4L
21. (a) f (x) = L 71.2 2_,
2+
n=1
2/.. ''''sen(n7rx/L)
(a) f (x) = 7r 1--,
ll
n=1
4
sen(2n - 1)x
2tt - 1
mt. x
cos -
3
cos1(2,z - 1)7r x1/..4
(2n - 1)2
597
598
1
1
2
1
(b) g(x) = - +
4
n7rx
4 cos(n7r/2) + 2n7r sen (n7r/2) - 4
COS
2 n 2
2
n
n=1
cos (n7r/2)nrrx
E 4 sen 2n7r
,2
2
5
12 cos n7r + 4
cos
(b) g(x) = - + E 2
cos n7r) 32(1 - cos n7r)
° n 7r (3
h(x)
E
n37r
c*
h(x) -
sen
n7
n,1
117T X
12
n=1
2
2
c
+5
+
=
n7rx
2
sen 3
n=1
6n27r2(2 cos nn - 5) + 324(1 - cos n:r)
30. (b) g(x) = 4
- +L
E [4
n42,4
cos
n=1
'ICY)
=
cos Tyr + 2
144 cos 1I7T + 180
n37r3
tur
x
3
127 X
sen3
40. (a) Estendaf(x) antissimetricamente a (L, 2L), ou seja, de modo que f(2L - x) = -f(x) para 0 < x < L.
Depois, estenda esta funcäo como tuna funcao par a (-2L. 0).
Seciio 10.5
2. X" - Ax X = 0, T' + AtT = 0
1. xX" - xX = 0, T' + AT = 0
4. [p(x)XT + XT(x)X = 0, T" + AT = 0
- X(X' + X) = 0, T' + AT = 0
3.
5. Näo 6 separavel
Y=0
6. X"+(x+A)X =0, Y"
u(x,t) = e -4IX1T2 'sen 27rx - e -2500x2f sen 57rx
sen 2:-rx
u(x, = 2e - " 2EI Nen (7rx/2) - e -n2 " 4 sen7rx +
1 - cos nz 2 2 1600
100
e
sen u(x, = —
r
40
/11
E
160 tsen(mr/2) _„2,20600
tur x
e
SC n
u(x , t) =
7r-40
n=1
- cos(3mr/4)
E cos(n7r/4)
u(x, t) =
E e-"""1/1("sen 40
7r
tt(x,t)
100
—
IT
,
e-n-n-01600
(— 1
)n+1
117 I X
40
n=1
80
n
lig
n =1
t = 5, n = 16; t = 20, n =
t = 80, n = 4
(d) t = 673,35
15. (d) t = 451.60
(d) t = 617,17
(b) t = 5, x = 33,20; t 10, x = 31,13; t = 20, x = 28,62;
t = 100, x = 21,95; t 200, x = 20,31
(e) t 524,81
200 cc 1 cos nn - n ,- n2a2 t 400
"X
10,
-
E
e
"
"
sen
t = 40, x = 25,73;
20
(a) 35,91°C
(b) 67,23'C
(c) 99,96°C
(a) 76,73 s
(b) 152,56 s
(c) 1093,36 s
(a) aw x., - bw, + (c - bt5)w = 0
(b) 8 = c b se b 0
tt
2
y
r a 2 x2 T = 0
±
(
x2
_
)
0,
=
X" + 11 2X
( x2 r2 _ 4 2 )R = 0,
9,
+
i.,29
23. r2 R" + rR' +
=0, T' + a 2 ), 2 T =0
Seca() 10.6
1. u = 10 + ix
3. u = 0
5. u = 0
7. it = T(1 + x)/(1 + L)
9. (a) u(x , = 3x + 2_,
n=1
(d) 160,29 s
2. u = 30 -4 r
4. u = T
6. u = T
8. =
+ L - x)/(1 + L)
70 cos n.7 + 50 _0.86n2,2000
nnx
sen
ri g
20
10. (a) f (x) = 2x, 0 < x < 50; f (x) = 200 - 2x, 50 < x < 100
+ E cne-1,14n2n2t/(100)2 sentrr—x
100
5
tur
40
800
cn = _ s en
tur
n2 a 22
oo; 3754 s
(d) u(50, t)
10 quando t
(b) u(x, t) = 20 -
11. (a) 1i(X ,
=
30 - x +
E
c„e-n2jT2I/9°3 sen T x
30
n.I
60
c,, = 3 3 [2(1 - cos na) - n2 a 2 (1 + cos rut)]
11 7
2
12. (a) u(x,t) = - + ECne-n 2 :r 2 a 2 " 2
cos
n=1
nrX
L
n impart
-4/(n2 - 1) 7 ,
n par
(b) lim,..„„ u(x,t) = 2/n
ti
200
13. ( a ) /1 (x, t) = —+ Ec„e-n 2 ' 2 " 6" cos t40
9
160
cn = (3 + cos na)
3,1272
c, =
0,
(c) 200/9
(d) 1543 s
-''
177 X
25
n2,20900
(a) 11(X , t) = — + E c„e-'
cos 30
6
n=1
50
rut
1177
Cn = — (sen — - sen —)
na
3
6
(2n - 1 urx
(b) u(x,t) = Ecne-'2"- n27t2.20L2 sen 2L
n=1
2 L
(2n - 1)7rx
dx
c„ = L f f (x)sen
L., 0
2L
(2n -6 )7 x
16. (a) 11(X , = Ec„e- (211-112.7r2t13W)serl
01
n=1
120
, [2 cos IITT ± (2n - 1)7]
=
(2n - 1)-u(c) x„, aumcnta a partir de x = 0 e chega a x = 30 quando t = 104.4.
(2n - 1)ax
cme.-(2"-1)2'2`/36wsen
17. (a) u(x,t) = 40 +
60
n=1
40
na - (2n - 1)7]
c„ =
(2n - 1)272[6 cos
E
0 < x < a,
a[ + (L la) - 1]
La < x < L,
T[1
a
(L1a) - 11
sen kt„x
(e) u,(x,t) =
v(0) = T1 , v(L) = T2
a 2 v" + s(x) = 0;
T
u(x) = I
onde = K2A2/xIA
w(0,t) = 0, w(L, t) = 0, w(x, 0) = f (x) - v(x)
w, = a2 w„;
(a) v(x) = TI + (T2 - Ti )(x L) + kLx 12 - kx 2 12
160 (cos to - 1)
c„e -"2 '2114°° sen nnxc„ =
(b) w (x, t) =
En =1
20
23. (a) v(x) = T1 + (T2 - TI )x/ L + kLx/6 - kx3I6L
(b) w(x, t) =
E
n=1
c„ =
c„e-'2#2`146° sen
177 X
20 '
20[ 3m 3a 3 (3 cos nur - 1) + 60 cos nut
m4n4
3
n3a3
599
Se* 10.7
nrrx
nit
8 x 1
cos
sen sen1.,(a) u(x, t) =
7. - n=i n
„ E -,
w rat
.
L
E,
wr x
wren
3wr
nit
8 x 1
- en L cos L
sen 4-+ sen4s
(a) u(x,t) = - ,
n4
4
n=1
rural
n7rx
32 ,--, 2 + cos n7
cos sen
(a) 10, t) = --i L 3
L
L
!I
7r- n=1
4 x sen(n7/2)sen(n7r/L) sen turx cos rural
(a) u(x,t) = L
L
n
E
rr
n=t
nit at
tur x
wr
8L 't•-.., 1
(a) Il(X, t) = - 2_, - sen 2- senL-senL
n3
(1;7 3
n=1
n7rx
brat
8L N sen(n7/4) +sen(3n7/4)
sen
sen
(a) 11(1,0= --=,
L
Ln3
cur' n=1
E
-4- E 32L x cos nir + 2
(a) tt(x,t) =
(ITT
n=
11
I
.1
sen
nit at
tur x
sen L
L
4L x sen(n7/2)sen(n7r/L)
E
(a) tt(x,t) = -7
TM'
n2
n=1
sen
at
mrx
senwr
L
L
(2n - 1)7 at
- 1)7rx
cos
2L
E c„ sen (2n 2L
u(x,0 =
n=1
2
(2n -
L
f f(x)sen
L. 0
c,,= 7
8 x (a) tt(x.t) = 71,
n=1
1)7rx
2L
1
sen 2n - 1
th
(211 - 1)7.(2n - 1)7r
st..n 4
2L
sen
(2n - 1)7rx
(2n - 1)7r at
cos
2 L
21.
(2n - 1)rr at
(2,1 - 1)7rx
(2n - 1)7 + 3 512
cos
ti7
x
cos
s,..11 11. (a) u(x.t) = 74- L
2L
2
(2n
1) 4
7 n=1
(b) (P(x + at) representa uma onda movendo-se no sentido negativo de x com velocidade a > 0.
(a) 248 ft/s (b) 49,67n rad/s (c) As freque.ncias aumentam: os modos permanecem inalterados.
+ IA 2 .0 = 0,
T" + A. 2 a= 1- =t)
21. r2 R" -1 rR' + (A.2 - ii 2 )R = 0.
(c) y = 0
(b) a„ = (1 3 1 + (y 2 L2 /11 2 7r 2 )
20 (
nit
2tur
(a) c„ =
2 sen
sen
sen 3'_17
7r
2
)
n2
2
5
Seca() 10.8
E
OG
(a) u(x,y) =
n=i
fir r x
I1TT V
cn sen- senh ' ,
a
a
„=
2/a
senh(n7b/a)
10
g(x)sen TX
a
dx
4a ,1 sen(n7/2)
nar y
IIIT X
(b) u(x,y) = -.-.,
- sena 4 n=1 n2senh(n7b/a)sen a
a
„,- E
E c sen -senh
he
u(x,y) =
117 X
n
a
n=1
njr(b - y)
a
= 2/ Cl
senh Our Nu )
,s,
fo
11(X)sen
E
X
- dx
a
tr. r x
n7(1) - y)
wry
3. (a) u(x,y) = Ecn(l) senh ,
c,,(2) sen
sen senh +
a
a
h
h
n=1
n=1
h
.
2/b
FUT X
2/(r
wr y
(1)
cn h(x)sen
eh
dy, (1,2) =
f (y)sen
senh(wr a/ b)
a
senh (tur b / a)
b
117T X
f
(b) cl,"
5. it(r, 0) =
-
2
c (2) =
n7r senh(n7a/6)' "
Co
+
E r-"(c„
2
IV3IT '3
cos nO + k„ sen 110);
n=t
a n 2:r
c„ = - f
f f (0) cos nO dO,
7
an f
k„ = -f (0)senne dO
7 0
f
(n 27 2 - 2) cos n:r + 2
senh (nrr b I a)
RfSPOSTAS DOS PROBL EMAS 601
6. (a) u(r,e) =
f
f(0)senne de
E en esen ne9 , c„ = —
ra"
2 0
n=1
4 1
(b) c„ =
cos mr
Ira"
u(r,O) =
n3
E c„r"'la sen ruare
n. re
re
(B)sen — de
c, = (21a)a'ia
n=1
DO
. (a) tt(x.y) =
2 f"
n7 r x
Ecn e -"Yia sen nrrx
c„ = f(x)sen—
a
a 0
a
n=1
4a2
(b) c„ =
(1 - cos nrr)
n- 7r 3
10. (h) u(x. y) = Co
(c) yo -=" 6,6315
wry
cos
E c„ cosh mrx
b b
c„ =
n=1
11.
u(r,6) = co +
2/nir
wry
f ( 1 . ) cos — dy
senh(nrralb) fo b
E e(c„ cos nO + k„sen ne),
n= I
1
=
nt-r a n
fo 2D g(0)sen ne de:
1
g(6) cos ne dO, k„ =
I f
o
2.7
a condicdo necessaria 6
(a) li(X y) =
(
g(0) de = 0.
r y
E c„sen —T Xcoshn7—,
a
= a
I
2/a
cosh(mrb/a)
a
g(x)sen X
a dx
sen(n:r!2)
(b) c„ =
n2 rr cosh(trr b / a)
4a
(a) u(x • y) =
E c„stnh (2ri 2b- 1prx sen (2n -2b1)rry
n=1
21 b
b
)
(2n 1:ry
dy
f(v)sen
2h
senh[(2n - )7ra 2h]
32h2
(h) c,, =
(2n - 1)37r 3senh[(2n - 1)7 ral2b]
=
.
coy
14.
(a) li(X, y) = —
2 +
0
E ,„
mix
cos
n=1
a
2
Co = —
a b
Lf
g(x)dx, c,, a 4
M y,
MT
— senn
sen h
a
a
2/a
,
a
senh(nrrb/a) fo
n:i X
g(x) cos a
dx
24a4(1 + cos mr)
n4 7 4 senh(mr b I a)
as 4aa w cos[(2n - 1)7.r/a] cosh[(2n - 1):rz/a]
16. (a) u(x, z) = b + -2— —r,2
(2i: - 1) 2 cosh[(2n — 1)7 rb I a]
1
n=
(b) Co = —
b
(1+
30)
•
C.'s -
E
C A PITULO 11
Secio 11.1
1. Homogenea
2. N5o homogenea
3. Nao homogenea
4. Homogenea
5. Ndo homogenea
6. Homogenea
7. (a) 0„(x) = sen n/;C x, onde
(b)
satisfaz f = - tan ./Tk 7r;
0,6204, A2 I" 2,7943
(c)
(d) A.„ '"=" (2n - 1) 2 /4 para n grande
(b) NA°
8. (a) 0„(x) = cos fix, onde
= cot NA-;
satisfaz
A l -24 0,7402, A2 ='"1-.- 11,7349
A.„ (n - 1) 2 7r 2 para n grande
9. (a) On (x) =sen,/,‘.7,x + 117, cos .„5;,x, onde ,A7; satisfaz
(X - 1)senJ - 2VA:cos
= 0;
(h) No
A l1,7071, A2 13,4924
X„ (n - 1) 2 1T 2 para n grande
10. (a) Para n = 1,2,3, , rp„(x) = senunx - cos 1. rr X e An =
onde tin satisfaz
= tan It.
(b) Sim: AO = 0.4(x) = 1 - x
602
. -59,6795
i 11:- -20,1907, A2 24
A„ z'A -(2n + 1)2 7 2 /4 para n grande
k 1
13. /2(x) = 1/x
15. p(x) = (1 -
12. ,a(x) =e-`2
14. p(x) =
X" + XX =0, T" + cT' + (k +
Xa 2 )T
X2)-1/2
=0
sentur x; n = 1, 2, 3, ..
(b)
= n2 7 2 , 4),(x) =
(a) s(x) =
satisfaz fn = (7)tan(3j.L); as autofuncOes associadas
onde
Os autovalores positivos sao X = =
0
6
urn autovalor corn autofuncao associada 0„(x) = xe-2-%
L
=
4,
A.0
sac. Ø„(x) = e-''sen(3,/r;x). Se
nao
existern autovalores negativos; se L > 1/2, existe urn auL
<
4,
=
0
nao
é
autovalor.
Se
#.
X
se L
tovalor itegativo X = -p 2, onde p e urna raiz de = (4)tanh(3pL); a autofuncao associada e 0_ 1 (x) =
e-2'senh(3px).
Niio tern autovalores reais.
= 0; a autofuncao associada é 0(x) = x — 1.
0 Cmico autovalor
cos J.. 0
-
(a) 2
-= 57,7075
X 1 -1- 18,2738, A2 1
cosh ,/Tz = 0, p =
2 senhji -
(e) A._ i L, -3,6673
24. (a) A, = itn, onde p„ e uma raiz de sen AL senhaL = 0, logo A„ = (n7/L)4;
A 1 97,409/L4 , X, -25. 1558,5/L4 , 0„(x) =sen(n7x/L)
onde pn e uma raiz de senpL cosh AL - cos L senh L = 0;
A.„ =
sen p„xsenh p„L - senp„Lsenh it„x
4,„ = A; -24 237,72/L4 , X 2 .1-- 2496,5/L4 ,
senh p„L
A.1 -1, 12,362/L4,
A, = it 4n , onde p„ é tuna raiz de 1 + cosh it L cos it L =
A2 485.52/L4
[(senp„x -senh p„x) (cos k(„L + cosh tin L) + (senp,L senh „ L) (cosh p„x - cos p„x)]
(x) -
cos kr„ L + cosh L
L=0
=sen VAT, x. onde A.„ satisfaz cos VX7, L - y,/Z Lsen
25. (c)
(d) A 1 1,1597/L 2 . A2 13,276/L2
Seciin 11.2
4sen(n - ;)7x; n =
2. Ø„(x) = 4 cos(n — ;)7i . x; n = 1,
cos fur x; n = 1, 2, ...
00 (x) = 1, 00 (x) =
1. 0„(x) =
4
4 cos -4 x
\FAT: = 0
k„ -
0„(x) = ;.,01/2 , onde A.„ satisfaz cos Ir
(1 +seri' „/—
5. 0„(x) = f e' senn7 r
n = 1, 2, ...
6. a„ =
2./2
= 1, 2, . .
(2n - 1)7
44 _
•
n =-- 1, 2, ...
(2n - 1)272
24
a„ = (1 cos[(2n - 1)7/41}; n = 1,2, ...
(2/1 - 1)7
2 N/72 s(ennin
i-)2z1)2(7/2);
a,, n = 1, 2, ...
an =
2
Nos Problemas de 10 a 13, a„ = (1 +sen2 fATT,) 1/2 e cos ,A7, -A7,sen,f/:, = 0.
a,, -
sen
X„
./ A na 11
;
n = 1,2, ...
11. n„ =
4(2
cos
- 1)
;
n = 1,2, ...
?••:(111
12. a„ - ; n = 1,2,...
13. a„ -
4sen(N/Z/2)
;
n = 1,2,...
n
14. Nao e autoadjunto.
15. Autoadjunto.
16. Nao e autoadjunto.
17. Autoadjunto.
18. Autoadjunto.
21. (a) Se a, = 0 ou b, = 0, nao existe o termo de fronteira correspondente.
25. (a) A l = 72/ L2 ; Ø1(x) sen(7xIL)
(b)"-=- " ( 4 , 4934 ) 2 / L2 ; 01(x) = se n
L
- ,/;:i x cos
(c) A. 1 = (27)2 /L2 ; 01(x) = 1 - cos(27x/L)
26. A. 1 = 7 2 /4/.2 ; cbi (x) = 1 - cos(mq2L)
(a) X" - (v/D)X' +i.X = 0, X(0) = 0, X'(L)= 0;
(e) c(x, t) = E a„e - '.' bre"l2D senAnx, onde ;t.„ =
ri =
603
T' +ADT = 0
(v2/4D2);
1
4Dp,2, f
r)senp„x dx
0
(2LDAn2 + rsen2 it „L)
(a) ri, +1;11, = Duxx , u(0,t)= 0, it,(L,1)= 0, tax, 0)
(b) 10,0
=
E
b„e - -- D è" i2D senit„ x, onde ;%.„ = p i::
-co
(v2/4D2);
o=1
I)„ =
8c0 ,0 2 p 2"
(2Dtt„e L12 9 cos tt „L + ve-'1- 12"sent t „L - 21)11n)
(v2 - 40 2 tt ,. 1)(2LIDI.q, + vsen 2 L)
Seciio 11.3
E
1. y = 2
(-1)"-Isenn7rx
2. y = 2 E
n 2 7 - 2)n7
I
y = 4
(-1rIsen(n - I)7x
2
,
[(n - i) 2 7r 2 — 2I(n - k)272
cos(2n - 1)7x
E
„=1 [(2n - 1) 2 7 2 - 21(2n - 1)272
4
5. y =8 E sen(n7/2)senn7.v
(n.27,
2_
2),i2,72
x
- 11cos
2)(1 - sen2 ,FA.7;)
cos
y = 2 E (2
u=
%.„ (;,„ —
6-9. Para calla problem. a solu45o é
y =
II=
"
On(X),
A n — it
cn
= f .fix)0„(x)dx,
0
to
0
onde o„(x) é dada nos Problemas de 1 a 4, respect ivamente. as Secio 11.2. e A„ c o autovalor associado. No
Problcma 8. o somatOrio comeca cm n = 0.
1
1
1
1 (
cos 7x + —
v=
- - + csen7rX
a =
2
27-
72
•
I?. a arbitrario. y = c cos 7.v + al72
N5o tern solucao
17. u(x) = a -(1) - a)x
13. a = 0. v = csen7x - (x/27) sen7x
V (X ) = 1 —
t t"
1
u(x.t) = N/5 [- 441 +
4c;
—
:r-
I.-
-
.
/7
2
(2n
=2
20. u(x.t) = 4
C„ = e
N/2
4c„
- 1)272
E [
./2- senj„
VT
.„ (1 ±sen 2
2
[1
e-°1-112
isen( 111.)7x,
an
•
Cn =
4 \(-1)'14-'
(2n - 1)272
n = 1,2, ...
cos 17
.„ x
(1 + seri X,)1/2'
e - '^ ` ) +
(e
;.„ _ 1
7A:
+ ---,, e- ' 2 1i's sen —
,
- cos .fl..;)
sen2VZ)li2.
= An (1
satisfaz cos j.; - ‘5.7,seniA.7, = 0.
r t(x,t) = 8 E
sen(n7/2)
„= I
u(x,t) = f E
e -"
717'
2 2
'r 1)
senturx
c„(e' - e"" -112)2'21 )sen(n - 2)7x
- 02,2 _
n= I
1
C„ =
2 .4(2t: - 1)7 + 4.(-1)"
(2n - 1)272
(a) r(x)w, = [p(x)w,j,-q(x)w, w(0,t) = 0, w(1,t) = 0, w(x,0) = f(x) - v(x)
4 '° e •- (2"- 1)2 ' 2 `sen(2n - 1)7x
Il(X,t) = .v 2 - 2x + 1 +
E
:r
11=
,1
2,r-1
25. u(x, t) = - cos 7x + e -9a'`"4 cos(37 v/2)
31-34. Em todos os casos a soluciio é y = f G(x, ․)f(s)ds.onde G(x, ․) é dado a seguir.
1
G(x, ․) =
x,
<S<X
_ s,
x<s<1
0< s < x
x<s<1
s(2 — x)/2,
G(x, ․) =
x(2 —s)/2,
I cos ssen (1 — x)/ cos 1,
sen (1 — s) cos x/ cos 1,
0<s<x
s,
G(x, ․) = x,
<s<1
0<s<x
G(x, ․)
Seciio 11.4
00
y=
E .
=1
x<s<1
1
f (-410(NA; X) dx / f 4 (fi.„ x) dx,
1
c„
A n — it
J0(N/T1 X),
C„ =
f
f
I
0
0:7, satisfaz .10 ( j) = 0.
00
cv „
(c) y = — ..1)- +
Jo(4, x);
An
— ii
A 4—'
n=1
I
1
1
co =
2 f f (x) dx; c„ = f
f f (410(1Z x) dx I f xfi,(Vr„ x)dx, n = 1.2, ... ;
0
0
,/Z satisfaz J,;(1i) = 0.
1
I
(d) a„ =ff xi k W A 72 x) f(x) dx 1 f x4(47, x) dx
.
0
c c.
(e) y =
,:
t
E c'i ikc,/7,
; Al. ,„ = f f (Oh ( 1k, -, x) dx /
n=1
An — A
0
,. ,
(b) y = cn p2„_,(0, c„ =
f co P2„_, 0,-) (Ix
—
A
0
n--- 1 X n
Seca() 11.5
(b)
+ 1), (1( , 0) = 0. 0 <
2) =
u(0, = u(2, ))) = 0. 0 < < 2
E k„Jo(X„r)senA„at, k,, =.
n=.1
0
I
I
7
u(r,t)
1 xf,(. ( ,/—
A n .x)
t (C IxPi„ _ 1 ( x ) dx
f
0
<2
f 1 r.10(Xnr)g(r)dr
r4(:.„r) d r
An a 0
3. Superponha a soluciio do Problem 2 c a do exemplo [Eq. (21)] no texto.
u(r, z) = Ecne- A "Vo(X,,r), (:„ = f rJo(A„r) f(r) dr I
rJ 2 (X„ r)
1
e X„ satisfaz Jo(X) = 0.
n=
(b) v(r , 0) = coJu(kr) E J„,(kr)(b„,sen me + c„, cos I11 0),
rn=1
b,„ =
1
(27r
ir. I ,n (kc) Jo
1
f (0) sen inO d0;
in = 1, 2, ...
2,
f (0) cos me d9; in = 0,1, 2, ...
1
1
8. c„ =ff rf (r)Jo(A,,r) dr I f r4(A„r) dr
o
00
1
1
10. u(p, ․) = E c„,on P„(s), onde c„ = f 1 f (arccos s)P„(s)ds / f Ps) ds:
cm = 7J„,(kc) f0
n=0
—I
P„ é o n-esimo polinOmio de Legendre c s= cos 0.
Seca() 11.6
1. n = 21
2. (a) b„, = (-1)"1+1 i21 nut
(c) n = 20
(c) n 1
3. (a) b,,, = 2../2(1 — cos in7)/111 3 7r 3
(a) fo(x ) = 1
(b) (x) = 0(1 — 2x)
—1 + 6x — 6x2)
(c) f2(x) ---,
(d) go(x) = 1,
g 1(x) = 2x — 1, g2 (x) = 6x2 — 6x + 1
P0( x ) = 1, PI(x ) = x, P2 (x) = (3x2 — 1)/2, P3(x) = (5x3 — 3x)/2

Respostas Livro Boyce-Diprima 9th Ed.

Transcrição

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