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Respostas dosProblemas

CAPiTULO 1 Seca() LI

1. y 3/2 quando t 2. y se afasta de 3/2 quando t oc3. y se afasta de -3/2 quando t x 4. y -0 -1/2 quando r -0 oo5. y se afasta de -1/2 quando r - cc 6. y se afasta de -2 quando t co7. y . 3 - y 8. y' = 2 - 3y9. y' = y - 2 10. y' = 3y - 1

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 valorinicial 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 ovalor inicial é menor do que 5.v = 0 6 solucäo de equilibrio; y 0 se o valor inicial c negativo; y se afasta de 0 se o valor inicialpositivo.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 valorinicial esta entre 0 e 2;y se afasta de 2 se o valor inicial d maior do que 2.(j) 16. (c) 17. (g) 18. (h) 19. (h) 20. (e)(a) dq/dr = 300(10 -2 - q10'):q cm g. r em h(h) q -> 104 g; naodl/ /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(a) nu,' = mg - kv 2(b) v ,/mg/k(c) k = 2/49y d assintOtico a t- 3 quando t -0 co 27. y 0 quando t -> 00y oc, 0 ou -cc, dependendo do valor inicial de yy -> cc ou -co, dependendo do valor inicial de yy co ou -oo ou y oscila, dependendo do valor inicial de yy -> -oc ou d assintOtico a -,./2t - 1, dependendo do valor inicial de yy 0 e então dcixa de existir depois de algum instante ti > 0y oo ou -00, dependendo do valor inicial de y

Secäo 1.2

I. (a) y = 5 + (yo - 5)e' (b) y = (5/2) + [yo - (5/2)]e-2'(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 depressaem (b) e (c) do que em (a).

2. (a) y = 5 + (yo - 5)ei (b) y = (5/2) + - (5/2)Je2'(c) y = 5 + (yo - 5)e21A solucäo de equilibrio é y = 5 em (a) e (c), y = 5/2 em (b); a soluctio se afasta do equilibrio mais de-pressa em (b) e (c) do que em (a).

555

q(0) = 5000 g (t) = 5000e-o3"'T = 300 In(25/6) 428,13 min 7.136 h

(b) q(d)

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.(a) ye = (b) Y' = aY(a) yi(t)=y = cc"' + (b/a)(a) T = 21n 18 -14 5.78 meses(c) po = 900(1 - e -6 ) 897,8

(a) r = (In 2)/30 dias-1(a) T = 51n 50 -= 19.56 s(a) duldt = 9,8, v(0) = 0(c) v *:-L, 76,68 m/s

(f) T Z.= 9,48 s

(c) T 24,5 (has1620 In(4/3)/ In 2 672,4 anos(a) u = T + ( 1 0 - T)e-kr6,69 h(a) Q(t) = CV (1 - e-oRc)(c) Q(t) = CV exp1-(t - ti)/RC]

18. (a) Q' = 3(1 - 10- 4 Q), Q(0) = 0

Q(t) = 104 (1 - e-300I ),t ern h: depois de 1 ano Q 9277.77 gQ' = -3Q/104 . Q(0) = 9277,77

(d) Q(t) = 9277,77e -"4 , t cm depois de 1(e) T 1-=., 2,60 anos

19. (a) q' = -q/300,(c) nao(e) r = 250 In(25/6) 356,78 gal/min

(b) y = cen + (b/a)

(b) T = 2 ln[900/(900 - po)] meses

(b) r = (ln 2)/Ndia-1(b) 718.34 m(b) T = .1300/4,9 7.82 s

(b) v = 49 tanh(t/5) m/s11. (e) x = 245 In cosh(t/5) m

(a) r 2.-4 0,02828 dia -112. (b) Q(t) = 100e-"2828'

(b) kr = In 2

(b) Q(t)-). CV =

ano Q 670,07 g

Seciio 1.3I. Segunda ordem, linear3. Quarta ordem, linear5. Segunda ordem, nao linear

15. r = -217. r = 2, -319. r -1, -221. Segunda ordem, linear23. Quarta ordem, linear

2. Segunda ordem, nil() linear4. Primeira ordem, nao linear6. Terceira ordem. linear

16. r = ±118. r = 0,1,220. r = 1.422. Segunda ordem, nil° linear24. Segunda ordem, nao linear

CAPiTULO 2 Seciio 2.1

(c) y = ce-3( + (03) - (1/9) + e -2`; y d assintOtica a //3 - 1/9 quando t(c) yc= e2t t3e2'/3; y se quando t oo(c) y = ce' + 1 +1 2 e-72; y 1 quando t cc(c) y (c/t) + (3 cos 20/4/ + (3 sen 20/2; y 6 assintOtica a (3 sen 20/2 quando t 00(c) y = ce2' - 3e'; y -> co ou -co quando t oo(c) y (c - t cost + sen t)/( 2 : v 0 quando t -> cc(c) y + ce -12 ; y 0 quando t co(c) y = (arctan t + c)/(1 4. -2,2) ; y 0 quando t oo(c) y = ce- 1 /2 + 3t - 6; y e assintOtica a 3t - 6 quando t -* co(c) y = -te" +ct; y oo, 0, ou -oo quando t co(c) y = ce' + sen 2t - 2 cos 2t; y e assintOtica a sen 2t - 2 cos 2t quando t oo(c) y = ce-`/2 + 3t 2 - 12t + 24; y 6 assintOtica a 31 2 - 12t + 24 quando t co

13. y = 3e` + 2(/ - 1)e2 ' 14. y (t2 - 1)e-27215. y = (3t 4 - 41 3 + 61 2 + 1)/121 216. y = (sen 01(217. y = (t + 2)e2' 18. y = 1.--2 [(7 2 / 4, _4+)1 - t cos t +sent]19. y = -(1 + 0e-7e, t A 0 20. y = (1- 1 + 2e-`)It, t A 0

-* 00

PESFOSTAS DOS PROBLEMS 557

(b) y = cos t + s sen t + (a + Deì2 ; no =(c) y oscila para a = ao(b) y = -3eti3 + (a + 3)e/2 ; ao = -3(c) y -oo para a = ao(b) y = [2 + a(37 + 4)e2"3 - 2e-'/2)/(37 + 4); ao = -2/(37 + 4)(c) y 0 para a = no(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(c) y -> quando t 0 para a = a()(b) y= (e' - e + a sen 1)/sen t; no = (e - 1)/sen 1(c) y 1 para a = ao

27. (1, y) = (1,364312;0,820082) 28. yo = -1.642876(b) y = 12 + A 788cos2t + sen 2t - 7,3- e -fict ; y oscila em torno de 12 quando t oo(c) r = 10,065778yo = -5/2

31. yo = -16/3; y -> -oo quando t -> oo para yo = -16/339. Veja o Problema 2. 40. Veja o Problema 4.41. Veja a Problema 6. 42. Veja o Problema 12.

Seclio 2.2

1. 3y 2 - 2x 3 = c; y¢ 0 2. 3y2 - 21n11 +x 3 1 = c; x A -1.y 5.1-- 0+ cos x = c sey 0 0; tambërn y = 0; em toda parte

3y. + y2 - X3 ± x = c; y 0 -3/22 tan 2y - 2x - sen 2x = c se cos 2y r= 0; tamb6m y = ±(2n + 1)7r/4 para todo inteiro

em toda partey =sen[ln lx1 + cl se x A 0 e <1; tambêm y = ±1

7. y2 - x 2 + 2(e - e -x ) = c; y + eY 0 0 8. 3y + y3 - x 3 = C; em toda parte(a) v = 1/(x2 - x - 6) (c) -2 < x < 3(a) y = - 2x2 + 4 (c) -1 < x < 2(a) v = 12(1 - x)et - 11 10 (c) -1,68 < x < 0,77 aproximadamente(a) r = 2/(1 - 21n0) (c) 0 < B <(a) y = -[2 Im1 + x 2 ) + 41 L2(c) -00 < X < 00

(a) y = [3 - 2 31 + 112 (c) lx1 < 415.

(a) y = + .14x2 - 15 (c) x >

(a) y = - i(X2 + 1)/2 (c) -cc <.x < 00

(a) y = 5/2 - 3X 3 - ex + 13/4 (c) -1,4445 < x < 4,6297 aproximadamente(a) y = - + - 8e - 8e- x (c) 1x1 < 2,0794 aproximadamente(a) y = Err - arcsen (3 cost x)1/3 (c) Ix - 7/21 < 0.6155

(a) y = [; (arcsen + 1]" (c) -1 < < 1

y3 - 3y2 - x - x3 + 2 = 0, lx1 < 1y3 - 4y - x3 = -1, 1X 3 - 11 < 16/3 ou -1.28 < x < 1,60y= -1/(x2 /2 + 2x - 1); x= -2y = -3/2 + ,/2.r - e + 13/4; x = In 2

25. y = -3/2 + jsen 2x + 1/4; x = 7r/4 26. y tan(r 2 + 2x); x = -(a) y seyo > 0; y = 0 seyo = 0; y- SeY0 < 0(1)) T = 3,29527(a) y -+ 4 ququando t -> oo (b) T = 2,84367(c) 3,6622 < yo < 4,4042

x = c y +

ad -,

bc In lay + + k; a 0 0, ay' + b 0

a a-(e) ly + 2x 1 3 IY - 2-r 1 = c 31. (b) arctan (y/x) - In lx1 = c

32. (b) X2 -1- y2 - cx3 = 0 33. (b) ly - x1 = cly + 3x1 5 ; tamba.1 y = x(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 = c38. (b) clx13 = 1y2 x2


Sec5o 2.3

1. t = 100In WO min Z-1' 460,5 min 2. Q(t) = 120y[l — exp(—t/60)]; 120y

3. Q 50e-0.2 (1 — e -02 ) lb 7,42 lbQ(t) = 200 + t — [100(200) 2 /(200 + 1) 2 ] lb, t < 300; c = 121/125 lb/gal;lira c 1 lb/gal

6;.505io e-riso + 25 _ 2051

-5 cos t + 5 ,2042 sen t(a) Q(t) =(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%(a) k(e" — 1)/r (b) $3930 (c) 9,77%k = $3086,64/ano; $1259,92 10. (a) 589.034,79 (b) $102.965,21(a) t 135,36 meses (b) $152.698.56(a) 0,00012097 ano- I(b) Qo exp(-0,000120970, t ern anos(c) 13.305 anosP = 201.977,31 — 1977,31e 0n2n , 0 < t < t1 6,6745 (semanas)(a) r a.- 2,9632; nilo (b) r = 101n 2 6,9315(c) r = 6,3805

15. (b) 0,83 16. t = In V /In min L- 6.07 min(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 hR 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)/rSuperior. T = 431 anos; Michigan. T = 71.4 anus; Erie. T = 6.05 anus; Ontario. T = 17,6 anos

20. (a) 50,408 m (b) 5,248 s 21. (a) 45.783 m ( b) 5.129 s(a) 48,562 m (b) 5,194 s(a) 176.7 ft/s (b) 1074,5 ft (c) 15 ft/s (d) 256,6 s(a) duldx = v (b) = (66/25) In 10 mi -1 :=4. 6.0788 mi(c) r = 900/(11 In 10) s L-' 35,533 s

o

nt 2g kvo m

) uo m k v(a) x, -- In 1 + — + t„, — In 1 + —)

k2 m g mg26. (a) v = —(mg/k) + Ivo + (mg/k)I exp( —kt/m) (b) r = — gt: sim

(c) v = 0 para t > 027. (a) vL = 2a 2g(p — (b) e = 4ira3g(p — p')/3E

(a) 11,58 m/s (b) 13,45 m (c) k > 0,2394 kg/s(a) v = Ri2g1(R + x) (0) 50,6 0

30. (b) x = ut cos A, y = —gt 2 12+ut sen A + h(d) —16L 2 /(u2 cost A) + L tan A + 3 > H

(e) 0,63 rad < A < 0,96 rad (f) u = 106.89 ft/s, A = 0,7954 rad31. (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

Seca() 2.4

1. 0 < t < 3

2. 0 < t < 43. 7r/2 < t < 37r/2

4. —oo < t < —25. —2 < t < 2

6. 1 < t <7. 2t + Sy > 0 011 2t + 5y < 0 8. 1 2 + y2 <19, 1 — (2 + y2 > 0 Ou 1 — t 2 + y2 < 0, r 0, y 0

10. Em todo o piano 11. y 0, v A 3

12. t tur para n 0, ±1, ±2 .... ; ye -1 13. y = ±iy,2) — 4t 2 se Yo A 0; It1 < IYoI/2

y = [(1/yo) — I se yo #0; y = 0 se yo = 0;o interval° 6 Iti < 1/VT se yo > 0; —oo < t < oo se yo 0

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 < coy -). 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 > 918. y --> -oc se yo < 0; y -> 0 se yo > 0y .- -oo se yo < y, ';',--' -0,019; caso contrario y 6 assintOtica a ,5-17

(a) Niio (b) Sim: faca t,,= 1/2 na Eq. (19) no texto(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)

1 1 i26. (a) y i (i) = -:

Y2( t )

= -- p(s)g(s)ds

AU) P0) ,,,

28. y = ±151/(2 + 5ct5 )J 1/2 29. y = r/(k + ere')y = ± EE / (a ± cee-2ff )JI/2

I 7

y = ± p 0I

) 2 p(s)ds + c . onde WO = exp(21 sent + 2 Tt)Ito

y = 1(1 - e -2') para 0 < t < 1; y = -!;(e2 - 1)e -2 ' para t > 1

y = CI para 0 < t < 1; y = e- ''''' para t > 1

Secii() 2.5

y = 0 ë. instavely = -alb 6 assintoticamente estiivel.y = 0 6 instavely= 1 e assintoticamente estavel, v = c v = 2 sao instaveisy= 0 6 instavel 5. y= 0 e assintoticamente estavel

6. y = 0 6 assintoticamente estavel 7. (c) y = [y„ (I - y„)kt]/[ I + (1 - y„)kt]y = 1 6 semiestavely -1 e assintoticamente estavel,y = 0 e semiestavel.y = 1 c instavely = -1 e y = 1 sac) assintoticamente estaveis,y = 0 6 instavely= 0 e assintoticamente estavel. y= b =la 2 6 instavely= 2 6 assintoticamente estavel, y= 0 6 scmiestavel.y = -2 6 instavely 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

17. (a) y = K exp{ [In(yo/K)]e-"I (b) y(2) -1' 0.7153K 57.6 x kg(c) r 2,215 anos

18. (b) (h/a),/ kla7r; sim (c) k /a < 7ra219. (b k2/2g(aa)2

(c) Y = Ey2 = KE[1 - (E/r)] (d) Y„, = Kr/4 para E = r/2(a) Y12 = K[1 - (4h/ rK) j/2

(a) y= 0 6 instavel,y = 1 e assintoticamente estavel( b) Y = Yof[Yo + (1 - yo)e't(a) y = yoe- 13 ' (c) xo exp(-ayo/f3)(b) x = .vo exp[-ay0 (1 - 131

(b) z = 1/[v + (1 - u)e13 '1 (c) 0.0927(a,b) a = 0: v = 0 6 semiestavel.a> 0: y= f e assintoticamente estavel e y=- instavel.(a) a < 0:y = 0 6 assintoticamente estavel.

a > 0: y = 0 é instavel; y = jti e y = sit() assintoticamente estaveis.(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) ' - ppat

(b) lira x(t) = p: x(t)pat +

L

560 RESPOSTAS DOS PROBLEMAS

Sectio 2.61. x2 + 3x + — 2y = c3. x3 — x2y + 2x + 2y3 +3y = c5. axe + 2bxy + cy2 = k7. c' sett y 2y cosx = c; tambem y = 09. e'Y cos 2x + x2 — 3y = c

I 1 . Niio 6 exatay = [v — 3x2 ]/2, Ix' < ,123-3/3

y [x — (24x3 + X2 - 8x — 16)1/2J/4,15. 13 = 3; x2y2 + 2x3y = c19. X2 2 In lyl — Y-2 = c; tamb6m y = 021. xy 2 — (y 2 — 2y + 2)eY = c

24. (t) = exp f R(t) dt, onde r = xy

26. it (x) ; y = ce + 1 + 27. it (y) = y; xy + y cos y — sett y = cR(y) e2Y /y; xe2 — In IA tambem y = 0u(y) = seny; e sen y + y2 = c 30. ,u(y) = y2; xs 3xy + y4 = c

31. p(x,y) = xy; x3y + 3x2 + y3 =

Seclio 2.7(a) 1,2; 1,39; 1,571; 1,7439 (b) 1.1975: 1,38549; 1.56491; 1,73658(c) 1.19631;1,38335;1.56200: 1,73308 (d) 1,19516; 1,38127; 1.55918; 1,72968(a) 1.1; 1,22; 1,364; 1,5368 (b) 1.105: 1,23205: 1,38578; 1,57179(c) 1,10775: 1,23873; 1,39793; 1,59144 (d) 1.1107; 1,24591; 1,41106; 1,61277(a) 1,25; 1,54; 1,878; 2.2736 (1)) 126; 1,5641; 1,92156: 2,34359(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; 0,8315710.277920; 0,501813; 0.678949; 0.815302

( d) 0,271428: 0,490897; 0,665142; 0.7997295. Converge para y 0; Mio esta definida para y < 0 6. Converge para y 0: diverge para y < 0

ConvergeConverge para ly(0)1 < 2,37 (aproximadamente); diverge nos out ros casos

9. Diverge 10. Diverge11. (a) 2,30800; 2,49006; 2,60023:2,66773; 2.70939; 2.73521

2,30167; 2,48263; 2.59352: 2.66227; 230519; 2,732092.29864; 2,47903; 259024; 2,65958; 2,70310: 2,73053

(c1) 2,29686; 2.47691: 2.58830; 2,65798; 2.70185; 2,7295912. (a) 1,70308: 3,06605; 2,44030; 1.77204; 1,37348; 1.11925

1,79548; 3,06051; 2,43292; 1,77807; 1,37795; 1.121911.84579; 3,05769; 2,42905; 1,78074;1,38017; 1,12328

(d) 1,87734; 3,05607; 2,42672; 1,78224; 1,38150: 1,1241113. (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; 1,4949014. (a) 0,950517; 0,687550; 0,369188; 0,145990; 0.0421429; 0.00872877

0,938298; 0,672145; 0,362640: 0.147659; 0,0454100; 0.01049310,932253; 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; —3,09810Uma estimativa razotivel para y cm t = 0,8 6 entre 5,5 e 6. I\15o e possivel obter uma estimativa confiavelem 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.(b) 2,37 < ao < 2,38 19. (b) 0.67 < ao < 0,68

2. Niio é exata4. x2y2 + 2xy = c6. Nilo 6 exata8. Não 6 exata

10. y In x + 3x2 — 2y = c12. X2 + y2 = c

x > 0.984616. b = 1: ezv + = c20. e sen y + 2y cos x = c22. x2 ex sen y = c

25. i.t(x) = e3.'; (3x2y + y3 )e3i = C

RE SBOSTAS DOS PROBLEMS 561

Secáo 2.8

1. dulds = (s + 1) 2 (u) + 2) 2 , w(0) = 0 2. dulds = 1 - (u) +3) 3 , w(0) = 02k lk

( a ) 0,i(1) =(c) C(/) = e2' - 1

k=1 •

" (-1)ktk

k!(a) 0„(t) - E (c) _1

k =1

(a ) C(t) = (-1) k +i (k+1 /(k 1)!2" (c) 11(11 n-c (Mt) = + 2t - 4Ek=1

tn+i

(a) .0„(t) = t(n + 1)!

0,(c) lim„_, „(t) = t

t2k 13k-1

7. ( a ) (P„(t) = E

k -11 . 3 . 5 . • • (2k - 1)

8. (a) c),(t) = E k=i

2 . 5 8 ... (3k - 1)

t3 t 3 (2 t3 (7 21 11 ( 15(a) ol (t) = 3; O2(t)

= S + 7 . 03(i) = 3 7 . 9 + 3 . 7 9 • 11 + (7 . 9) 2 • 15t4 14 31 3110

(13

(a) 0 1 (0 = 02 0) = t -01 ( )4 • 1 " =

4 + 4 • 7 16. 10 4- 64 . 13

-1

11. (a) 01 (0 = t. 02(t ) = t - t

+ - + 0(t8),

7t 5 14/6- + -I- 007).

5! 6!

3116- 00)

6!

t 2 1 3 t 4tst°02 (t) = - I -

2- +

6- -F

54 24- - - + 0(t7),

t 1t4 3/ 5 4t6

03(1) = -2 -

12 20 45- - - 0(17),

t 2t4 7t504 (0 = -t - -

2 + -

8 -

60 + -

15 + 0(17)

Seciio 2.9

y„ = (- (0,9)"yo; yn 0 quandon -> co= yo/(n + 1): y„ -+ 0 quando co

y„ = yo,/(n + 2)(n + 1)/2; y„ pc, quando n

y„ Y°' se n= 4kou n= 4k-1:

-yo, se 11 = 4k - 2 ou n = 4k - 3;y„ nao tem limite quando n coy„ = (0,5)" (y0 - 12) + 12; y„ 12 quando ny,, = (-1)"(0.5)"(yo - 4) + 4; y„ -> 4 quando n

7. 7,25% S. $2283.63$258,14(a) $804,62 (b) $877,57 (c) $1028,6130 anos: $804,62/rnès; $289.663,20 total 20 anos: S899,73/mês;$215.935,20 total$103.624,62 13. 9,73%

16. (b) u„ -co quando n co19. (a) 4,7263 (b) 1,223%, (c) 3,5643 (e) 3,5699

Probleinas Variados1. y = (c/x2 ) + (x3/5)3. x2 + xy - 3y - y3 = 05. x2y + xy2 + = c

7. x4 + x - y2 - y3 = c9. x2y + x + y2 = c

2. 2y+cosy -x- senx=c4. y = -3 + cex-'6. y = (1 - el-x)8. y = + cos 2 - cos x)/x2

10. x+ + x -I + y -21n = c; tambt3rn y = 0

t 2 t3 t03(0 = t - 2!

+3!

+

,t- 3t -7t'04 = t - -

2!-3!

-

t312. (a) 0 1 0) = -t - t- -


23. v3t

= - + C•

25. (x'-/y) + arctan(1y/x) = c27. (x2 + y2 + OCT' = c29. arctan(y/x) - In NA2 + 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

(b) u = [b f 12(t) dt + c]/ µ(t), (I) = exp[-(at 2 /2) - bt)

36. v = + c2 + In t 37. y = In t + c2 +1y = (1/k) In 1(k - t)/(k +1)1+ c2 se c, = > 0; v = (2/k) arctan(t/k) + c2 se

-k2 < y = + c2 se c, = 0; tambem v cy = ±1., (t - 2c1)0 C/ c2 ; tamhem y = c Stigestrio: /1(v) = um (ator integrante.y + c2 - 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

42. y- = c, + 43. y = sen(/ + c2 ) = sen t + k, cost

44. 1.13 - 2c l y + c2 = 2t; tambem y = c 45. t + c, = (y - 2c1)(Y co1/2

46. ylnlyl - y + c i y + t = c2 ; tambern y = c 47. e = ( t + c2) 2 + c148. y = + 1)312 - 1 49. y = 2(1 - 0-2

y = 3 In t - In(t 2 + I) - 5 arctan t + 2 + ; In 2 +y=;t2 + 2.

CAPITULO 3 Secao 3.1

1. y = c i e + c2 e -3' 2. y = c l e - ' + c2e-2:3. v = c i et/2 + c2 e- ' 13 4. y = c l e` 12 + c2e'5. y = c l + c2 e -5' 6. y = c l e302 + c2 e-31 i 2

y = c 1 exp[(9 + 3.J)t/2] + c2 exp[(9 - 31-5-)t/2]y = c 1 expR 1 + 0)(1+ c2 expl(l - 173)t] 9. y = e': y -> oo quando t -> co

10. y = ;-e- ' - 1 e - 3; y -> 0 quando t -> cc 11. v = 12e0 - 8e02 ; y -> - cc quando t -> 00y = -1 - e- 3 '; y -> - 1 quando t -> ccy = A (13 + 5..ii) expR -5 + ../13)t/21+ A ( 13 - 5../i3) exp[( -5 - i13)t/2]; y -> 0 quando

DO

y = 33) exp[(-1 + - (2/./3) exp[(-1 - N/33)//4]; y quando t -> coy = y -> co quando t -> ooy = 10+20 + ;6,-(r+2)/2; y -> -oo quando t -> ccy" + y' - 6y = 0 18. 2y" + 5y' + 2y = ()y = et + e-1 : o minim° y = 1 em t = 1n 2y = -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 > 1y 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.(b) 0,71548 quando t = s In 6 "=- 0,71670 (c) = 2(a) y (6 + t3)e-2` - (4 + /3) e -"

t,,, = In[(12 + 3/0/(12 + 2/3)1, y„, = + 0)3 /(4 )4)2

p = 6(1 + L- 16,3923 (d) t„, -> In(3/2), y,,, -> oo(a)y = d/c (b) aY" +bY' + cY 0(a) b > 0 e 0 < c < b2 /4a (b)c < 0 (c) b < 0 c 0 <c < b2/4a

11. (x3 /3) + xy + = c13. y = tan(.r + x2 + c)15. y = c/cosh2(x/2)

17. y = ce3x - e2"

19. 2xy + xy3 - = c21. 2xy2 + 3x2y - 4x + y3 = c

e2t

12. y = ce' + e- ` +14. x2 + 2xy + 2y2 = 3416. e -x cosy + e2Y sen = c

x18. y = e -2" f e'

, cis +

20. e + CY =C

22. y3 + 3y - x3 + 3x = 2

24. sen ysen2 x = c

26. CYR + In 1x1 c

28. x3 + x2y = c30. (y2 /x3) + (y/x2 ) = c

132. - = -x f

S ds + 2

Y (b) y = r' + 2t(c - (2)-I


Seca() 3.2

1. 2. 13. e -4 ( 4. x2ex

5. -e21 6. 07. 0 < t < oc 8. -oc < t < 19. 0 < t < 4 10. 0 < t < oc

11. 0 < x < 3 12. 2 < x < 3:r/214. A equacao é nao linear. 15. A equacao é nao homogénea.16. Nao 17. 31e 2: +ce2'18. te` + ct 19. 51,1/(f ,g)

-4(t cost - sent)v3 e y, formam urn conjunto fundamental de solucties se e somente se a,b 2 - a,b, � 0.

-2 Iy i (t) = ie' y 2 (t) = t + jet

v1(t) = -1e -311-1) + Y2(t) = - .12 -r - e4-3"-1) ' 1 -11-1)

24. Sim 25. Sim26. Sim 27. Sim

(b) Sim. (c) iy,(t).y.,(t)] e [y,(t),y,(t)] sac) con juntos fundamentais de solucOes; ry2(t),y3(1)] e b74(t),y,(t)]nao saoct 2 e' 30. c cost

31. c/x 32. c/(1 - x2)34. 2/25 35. 3 "=- 4.94636. p(t) = 0 para todo t40. 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-•212Ic'2 dt + c2e 0/2

xu

Nao' (I) , , * 1 cos x ,

Sim. y = ILc; f 1.1- at m cl it (X) = exp [- j (-

v + -) ad

12(x) tAo x-Sim. y = c l x -1 + c2x 47. .r 2 u" + 3.r Et' + ( 1 +x 2 - v 2 ) 1. = 0

48. (I - x2 )p" - 2x/1' + a(a + 1)/./ = 0 49. /1" - xu. = 051. As equacOes de Legendre e de Airy sao autoadjuntas.

Sec:10 3.31. e cos 2 + ie sen 2 -1,1312 + 2,4717i 2. e2 cos 3 - ie 2 sen 3 -.4_ -7,3151 - 1,0427i

-1e2 cos(g /2) - ie2 sen( g/2) -e2 i -7,389112 cos(In 2) - 2isen (In 2) 1,5385 - 1,27791r l cos(2In>r)+ i;r -t sen (2 In ir) -0.20957 + 0.239591

7. = clet cos t + c2 e`sent 8. y = c l ef cos + c2 esen t9. y = c l e21 + c2 C4' 10. y = c l e' cos t + c2 e -i sen t

11. y = c l e -31 cos 2t + c2 e -3r sen 2t 12. y = c 1 cos(3t/2) + c2sen(3t/2)13. y = cos(t/2) + c2 e- ' son (02) 14. y = c i e(13 + c2e-40315. y = c l e -112 cos t + c2e'a sent 16. y = cos(3t/2) + c2e-2rsen(3t/2)17. y = sen 2t; oscilacáo regular

y = e 2r cost 2e -21 sen t; oscilacao decrescentey = -e'-'T/ 2 sen 2t; oscilacao crescentey = (1 + 20) cos t - (2 - 0) sen t; oscilacao regulary = 3e-'12 cos t + ;e- '' 22 sen t; oscilacao decrescentey = ../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 ( fn 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

25. (a) y =2e- ' cos ../5 t + [(a + 2)/A c' sen ./3 t (b) a = 1,50878

(c) t = (7r - arctanI20/(2 + a)))/,/3 (d)26. (a) y = e' cos t + ae-ai sen t (b) T = 1,8763

(c) a = a, T = 7,4284; a = 2, T = 4,3003; a = 2, T = 1,5116

L._


36. y = c 1 t -1 + c2t-238. y = c i t6 + c,t-140. y = cl t cos(2 In t) + c2 1 sen (2 In t)42. y = c i t -3 cos(In t) + c2 t -3 sen (In t)

35. y = c 1 cos(In t) + c2sen(Int)37. y = c 1 t -1 cos(; In t) + ot -1 sen (; In t)39. y = c i t 2 + c,t341. y = c i t + c2t-3

Sim, y = c i cos x + c2 sell x, x= f e-'212 dr

Niio46. Sim. y = c l e-'214 cos(ij t 2 i4) + c2 e-'214 sen (i3 t2/4)

Seciio 3.42. y = + c2 te- ' 31. y = c l e + c2te4. y = ci e -3" + c2 te -3"3. r = c i e- ` 12 + c2e3'I26. y = c l e 3' + c2 te3'5. y c i e cos 3t + c2 esen 3t

7. y = c i e-0 + c2 e -11 8. y = c l e-3r;4 c2te-3'''10. y = Cl/2 cos(t/2) + c2 e-ti2 sen (t/2)9. y = c l e2'15 + c2te2t/5

y = 2e20 - 3le2r/3 , y -co quando t ooy = 2te31 , y co quando t ooy = -e -113 cos 3t + 9e-ti3 sen 3t, y -› 0 quando t --> 00y = 7e -20' 1 ' + 5te-2('+'), y 0 quando t -› co

15. (a) y = e-3'I2 - te -3(/2 (b) t = -25to = 16/15, yo = -1- -0,33649y = (b + 4)te -302 ; h = -;

16. y = 2e02 + (b - 1)tel2-; b= I17. (a) y = e-'12 + ite-'I2 (b) t it = 5, ym = 5e- 3.5 2.24664

y = P -4I2 (b + 1)te-t/21M! = 4b/(1 + 2b) -4 . 2 quando b -+ co; y m = (1 + 2b) exp[ -2b/(1 + 2b)] 00

quando b -> co18. (a) Y = ae-21/3 -F (ia - 1)te-2'1323. y2 (t) = t325. y2 (t) = t- I In r27. y2 (x) = cosx229. y2 (x) = x' 'e-2ìi

Y32. y = c l e-' ,2x:' ease/2 ds + c2 e -ix2/2f

o34. y2 (t) = t- 1 In t36. y2 (x) = x39. (b) Yo + (a/ b))/042. y = c i t -1/2 + c2 t -112 In t44. y = c i t -1 + c2 t-- 1 In t46. V = c, t -2 cos(3 In t) + c2 t -2 sen (3 In I)

( b) a =24. y2 (t) = t•-226. y2 (t) = re'28. y2 (x) = x30. y2 (x) = x -1/2 cos .v

33. y2 (t) = y i (t) f yi-2 (s) exp [- f p(r) d rids(0

35. y2 (t) = cos t-37. y2 (x) = x -112 cos x41. y = c,t2 + c2 t2 In t43. y = c i t + C451245. y = c, 1 3/2 C2t3/2 In r

Secao 3.5y = c l e3( +c2 e- ' - e2r

y = c l e- ' cos 2t + c2 e -t sen 2t +sen 2t - 11 cos 2t3. y = c i e3' + c2 e -' + + 4. y = c 1 + c,e-2' + t - sen 2t - cos 2t

y = ci cos 3t + c2 sen 3t + (9t2 - 6t + 1)e3` +y = + c2 te - ' + t2e-'y = c i e-1 + c2 e-'/2 +12 - 6t + 14- sen t - cos ty ci cos t + c2 sen t - It cos 2t - sen 2tu = c l cos wot + c2 sen coot + (4 - (02 ) -1 cos Cotu = c i cos wot + c2 sen coot + (1/2w0 )t sen cooty = cos(il3 t/2) + c,e ' I2 sen (../T3 t/2) +b e' -y = Cie-1

c.2e2t + 13. y = et - l e -2r t -414. y L' cos 2t + 14 2 - 1 SC 15. y = 4te - 3e + r 3 ei + 4

16. y = + - - re,-' 17. y =- 2 cos 2t - 18- sen 2t - it cos 2t

y = e-' cos 2t + sen 2t + te-' sen 2t(a) Y (t) = t(A 0 t4 + 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


(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 + E l )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 + t 2 (Bot + B1)e2i + (Dot + D 1 ) sen 2t + (Eot + E I ) 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 + (80 1 2 + 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 + E 1 )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,E l = 1/25(b) to = + (-le

28. y = c 1 cos At + c, sen At + E ta„,/(A2 — m272 )1 sen mar trn.I

I t. 0 < t < 729 " Y = 1 -(1 + 7/2) sen t - (7/2) cos t + (7/2)e', t > 730. y = I 1 - e-•-: sen 2t - 1e - ' cos 2t, 0 < t < 7/2

Seca° 3.61. Y(t) = 2. Y (r) =3. Y (r) = ,1r 2 e -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 ty = c 1 cos 2t c2 sen 2t + (sen2r) In sen 2t - it cos 2ty = 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

= c l e2' + c,e3' + f [e31-s) - e2(`-s)]g(s)ds

Y= c 1 cos 2t + c2 sen 2t + f I sen 2(t - s)1g(s) ds

13. Y(t) = ; + f 2 In t 14. Y(t) = -2/ 215. Y(t) = 1(t - 1)e2t16. Y(t) = -2(2t - 1)e-117. Y(.v) = 1. x2 (In .0 318. Y(x) = - ix112 cos x

19. Y(x) = f (A:1et- r)--ffei' g(r) ell 20. Y(x) = X -1/2 f t -3/2 sen(x - t)g(t) di(

(b) y= yo cos t + y', sen t + sen (t - s)g(s)dstofi( .

y = (6 - a) I f [e'''') - ea('-')]g(s) ds 25. y = it -1 14 e(` -'5) sen µ(t - s)g(s)dsto fii

i29. y = c i t + c2 t2 + 41 2 In t26. y= f (t - s)eau ' 3 g(s) dsf

y = c 1 t -I + C2r5 + i4i 31. y = c 1 (1 + t) + c2 e + 1 (t - 1)032. y = c1 e` + c2 1 - ;(2r - 1)e-`

-1(1 + 2 )e - ' cos 2t - (1 + e `12 )e - ' sen 2t , t > 7/2Niio 34. y + c2 e -` -

11.

12.


Seca() 3.7It = 5 cos(2t - 6). S = arctan(4/3) 0,9273u = 2 cos(t - 27r/3)u = 2./3 cos(3t - S), S = -aretan(1/2) -0,4636u siff cos(7rt - 6). S = n + arctan(3/2) 4,1244u cos 8t ft, t ern s: = 8 rad/s, T = 7r/4 s, R = 1/4 ft

u ; sen 141 cm. t ems: t = 7r/14 s

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) 2.0113Q = 10 -6 cos 20001 C, t em su = 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 su (1/8,/31)e-2: sen (2 t) ft. terns; t = 7r. /2./31 s 1" :=-= 1.5927 sttl' 0,057198e-m5' cos(3.87008 t - 0,50709) m, t ern s; = 3.87008 rad/s,ttlak, = 3,87008/ ./175- 0.99925

Q = 10-6 (2e-5mr - 6.-1°°c ') C; t em s13. 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, ...

y = 8 lb•s/ft 18. R = 103 C2

20. vo < --ytt0/2m 22. 2n/ 31

23. y = 5 lb . s/ft 24. k = 6, r = ±2,is25. (a) r 41,715 (d) 1.73, min r 4,87

(e) r = (2/y) In(400/, 4 - y2)

26. (a) 11(t) = [110,4km - y 2 cos At + (2mvo + ytto)sen 11 I] /V 41:11? - y-

(b) =4m(ku,i+ yuor, + Ittu,)1( 411n - y2)

plu" + pogu = 0, T = 27 3pll pog

(a) u = f sen f t (e) horario(a) = (16/J23)c" sen(11727 t/8) (c)(b) u = cos( N klzt) - b v611-17: sen( 17fiTI )

32. (b) u = sen t, A 1. T = 27 (c) A 0,98, 7' = 6.07(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() 3.81. -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 sit" + lOu' + 98u = 2 sen (r/2), u(0) = 0, u'(0) = 0,03, u Cm in, t em s(a) u cos 16t + cos 3t (c) w = 16 rad/s

(a) u = 153!281[160e75' cos( 73 t) + e"-` sen t) - 160 cos(t/2) +

3128 sen(t/2)](b) Os dots primeiros termos sao transientes. (d) co= 4./3- rad/srt = f="1-

(cos 7t '4258 sencos 8t) = sen (t/2) sen (1502) ft, I erns45

It= (cos 8t + sen 8t - 8t cos 80/4 ft, t em s; 1/8, 7r/8, 7r/4, 37r/8 s(a) (30 cos 2t + sen 2t) ft, t em s (b) m = 4 slugs'

u = (12/6) cos(3t - 37/4) nt, tent sFo(t - sett 0, 0 < t <

15. 11 = Fo[(27r - t) - 3 sett tj, n < < 27r-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 acelera1 pc por segundo ao quadrado quando sob a acao do uma forca de 1 libra: 1 slug= 14.5939 kg. (N.T.)


10-6(e-4"' - + 3) C, t ems, Q(0,001) 1,5468 x 10-6;Q(0.01) 2,9998 x 10 -6 : Q(t) -* 3 x 10-6 quando t 00

(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 (w 2 - 1)(a) tt = [(co2 + 2) cos t - 3 cos cot]1(w2 - 1) + sen t

CAPITUL 0 4 Seciio 4.1

21. -oo < t < co . t > 0 ou t < 03. t > 1, ou < t < 1. ou t < 0 4. 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 independenteLinearmente dependente:f, (t) + 3f.(t)- 2f,(t) = 0Linearmente dependente: 2f,(t) + 13f2(t) - 3f,(t) - 7 f,(t) = 0Linearmente independente 11. 1

12. 1 13. -6e-2'14. e -2( 15. 6x16. 6/x 17. sen2 t = 10 (5) - z cos 2t19. (a) ao[n(n - 1)(n - 2) • • 11+ a l Inut - 1) • • • 2]t + • • + a,,t"

(aor" +a1rr.- • • a,,)e're' ,e2' sim. e` e' e2 ' ) 0, -cc < t < co

21. W(t) = ce -2 ' 22. W(t) =23. W(t) = c / t 2 24. W(t) = •/t27. y = ()el + c2 t + este' 28. y = c I t- + c,t 3 + c3 (1 +1)

Seciio 4.21 . f esi(:r14)+2.1

3. 3ei(T V2m.7) 4. erli3T1214.2nrri

5. 2eilIIIN/61+2rill 6. ,,./ed(sx/4,1 4 2/44:T/

7. 1. 1 (-1 + 1,( -1 - 8. 2 1 r4 e -Mi/S , 21/4eri/8

9. 1, -1. -i (./73 + - (./.7; + i) I Ni:5:

11. y = + c2 re' + y = c 1 e' + c2 ie' + c3t2e'13. y = c l er + c,e2' + 14. y = c 1 + (7,1 + c 3 e2z + c,te2'15. y = 1 cos t + sen t + e " = (c3 cos t + sen + e -15112 (cs cos !7 t + ct, sen t)

y = c, e: + + c3 e2' + c4e2'y = c 1 ei + c2 te' + c3t : e' + c4 e - ' + + c6r2e-'y = c i + c2 t + c 3 e' + c4 e -: + c5 cos t + c8 sen ty = c + c2 e' + c3 e2r - 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 tiy = (c 1 + c2 t) cos t + (c; + cot) sen t 23. y = c l e f + c2e(2+`f ± c3e(2-.3),

24. y = + C2e(-2+, c3e(-2- 32)(

y = /2 + c2 e -" 3 cos(t/0) + c3 c -r:3 sen (t/O)y = c 1 e3' + cze- 2: + + c4e(3-A`y = c I C° + c2 e- ' 14 c3 e - ' cos 2t + c4e - ' sen 2ty = c l e - ' cos t + c2 e -: sen t + c3c2' cos(0 t) + c4 e-21 sen t)

29. y = 2 -2 cos t + sen t 30. y = :15 sen(t/./2) - es/12 sen (t1 N/2)31. y = 2t - 3 32. y = 2 cos t - sen t33. y = le' - - - e--z/ 234. y = + P,e'12 cos t + e1/2 sen t

y 8 - 18e-'13 + 8e-r2170- ,

y = 2i; cos t - sen t - COS(39

+ sen(f t)

y = (cosh t - cos t) + (senh r - sen t)(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


Sec5° 4.3I. y e ' er + c2 tel + c3 e-' + Ite" + 3

y c l e` + c2 e-' + c3 cos t + c4 sen t - 3t - sen t

y = + C2 COS t + C3 SCI1 t + 4(t - 1)y = c i + c2 e' + c3e" + cos ty = + c2 t + c3 e-2( + c4e2i _ 3e'- t4

y = c, cos t + c2 sen t + c3 t cos t c4 t sen + 3 + ycos2t

y = + c2 t + c3 1 2 + c4e - ' + e' 12 [c5 cos(0 t/2) + c6 sen(0 t/2)] + .; t4y = + c2 t + c3 t 2 + c4e' + sen 2t + 16 cos 2ty = cos 2t) +

10. y = ( 1 - 4) cos t -(Zt+ 4) sen t + 3t + 4I1. y = I + 1(t 2 + 3t) - tei

49)y = - I cos t - sen t + + + e3' N cos 2t - 1. sen 2t

Y(t) = t(A0t3 -F A l t2 + A2t + A 3 ) + Bt2e'Y(t)= t(Aot + AOC' + B cos t + C sen tY(t) = At 2 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 + C I ) sent

Y(t) = Ae` + (Bot + + te' (C cos t + Dsen t)ko = a0, k„ = aoa" + + • • • + a„_ l a + an

Seciio 4.4

y = + c2 cost + c3 sen t - In cos t - (sen t) In(sec t + tan 1)

y = c, + c2 el + c3e-' - 1; 1 2 3. y = c i e! + c,e" + c 3 e2' +y = + c2 cos t c3 sen t + In(sec t + tan t) - t cos t + (sent) In cos ty = c l efc2 cos t + c3 sen t- 1, e-' cosy= c 1 cos t + c2 sen t + c3 t cos t + c4 t sen t - ll t2 sen ty c i e' + c2 cos t + sen t - (cos t) In cos t + (sen t) In cos t - cos t

- 2t sen t + -I e l2 / Coss I ds

8. y = + c3e-r - In sen t + In(cos t + 1) + f (e' 1 sen s) els-

17.

+ le" f (e'/ sen s) ds

c l = 0, c2 = 2, c3 = 1 em resposta ao Problema 4c 1 = 2, c, = c3 = c4 = em resposta ao Problema 6c 1 = Z. c, = Z, C3 = , to = 0 em resposta ao Problema 7c, = 3, c2 = 0, c3 = -e ra , to = :112 em resposta ao Problema 8Y(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) dsto

Y(t) f e(-" (t - s) 2g(s)ds; Y(t) = -tet In 111

Y(x) = z f [(x / t 2 ) - 2(x2 /t3 ) + (x3 It4 ) ] ,g(t) drxo

CAPITULO 5 Seca() 5.11. p= 13. p = oo5. p= i7. p = 3

f_ltx2n+1

9. E "

(2n + I)! P = 00n.0

2. p = 24. p= 2

6. p = 18. p = e

10. E; , =!'n=o n

RESPOSTAS DOS PROBLEMS 569

11. 1+ (x - 1), p = 0ooc

13. Ec_i>"÷i 1)" p = 1n=1ti

15. Ex", p = 1

n=0

12. 1 - 2(x + 1) + (x + 1) 2 , p = oo

14. E(-0"x", p = 1

n.0

16. E(-1)"+ 1 (, _ 2)", p =1n=0

= 1+22x +3 2x2 +42x3±... _ 02xn

y" = 22 + 32 •2x + 42 • 3x2 + 5 2 • 4x3 + • • + (n + 2) 2 (n 1)x" + • • •

y' = a l + 2a,x + 3a 3x2 + 4a4x3 + • • • + (n + 1)a„.,.ix" + • • •

= Ena„x"- 1 = E(n +1)a,:xnn= I 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„_:ann -2 n =0

ti

E21. E(n + 2)(n + 1)a,,.,2x" 22. a„-- 2x"n=0 n=2

23. E(n + 1)anx" 24. E [(n + 2)(n +1)an+2 - n(n - 1)a„n=0 n=0

25. E [(n + 2)(n + 1)(4,1.2 + ?lad? 26. a l + E +1)an.1. 1 + a„ - 1 ]•r"n=0 n=1

27. [(n +1)fla„ 4. 1 + a„ Ix" 28. a„ = (-2)"aoln!, n = 1,2 • a0e-2'

n=0

Seciio 5.2I. (a) a„ +2 = a„/(n + 2)(n + 1)

x2 x4 x6 xx:n(b,d) (x) = 1 + + + • • • = E

(211)1n=Ux3 x•

) . 2 (x) = x + 3!- +

51- +

71- + = E (2n 4. 1)! = senh

5 x 7 Y2"4-I

2. (a) (4.4-2 = + 2)x2 .r6 x

2n(b.d) y = 1 + - + +

=2 2 4 2 . 4 • 6 z+' 2nn1n=o

x 3 2nn!x?"÷Yz(x) = -3- + + 3 5. 7 (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 = 0k 2x4k4.0 k6xi2

(b,d) y i (x) = 13 -4 + 3 . 4 . 7 . 8 3 . 4 . 7 . 8- 11 • 12 +

m=0

Sugestlio: alga n = 4n1 n a relacäo de recorrencia.M = 1, 2, 3, ...5. (a) (n + 2)(n + 1 ) ani-2 - n(n + 1)an.,. 1 + a„ = 0, n > 1; a2 = -la()

(b) y i (x) = 1 - -1-x 2 - .1x3 - + • . • , y2 (x) = x - i6 x3 -- + • • •

= cosh X

cc: (_1)ni--1(k2x4)nt-I

1+ 34 7 8 4 34)nt=

k 2 X5 k4x9 k6x138 . 9 55+ 4 4 • 5 • 8 • 9 • 12 • 13

L [1 + E (-1)'"-3(k2x4)'•'4 • 5 • 8 • 9 . • • (4n1 + 4)(4m + 5)


(a) a„. f. 2 = —(n 2 — 2n + 4)an/i2(n + 1)(n +2)1. n 2; a2 = — (10. a 3 = —

(b) y i (x) 1 — .v2 + — :46 x6 + • • • .

Y2(x ) = x — 4 Xi+ 160x5 — X7 + • • •

(a) an+2-= —an/(n + 1),.v2

(b.d) y i (x) = I — 41

n= 0.1.2....x4 X 6

. =+ (-1)nx2n1. 3 1 .) • =1 1 • 3 • 5 • • • (2n — 1)

X3 x5 x. (_1)nx2n+1

Y2 (X) = X - + 2 . 44 - 2 • 4 6 + = n=} 2 4 • 6 • • • (210

(a) (4.4.2 = — [ (n + 1) 2an+1 + a„ + an.-1]1( 11 1 )(ri + 2), n = 1,2....a2 = —(ao + a l )/2

(b) y i (x) I — 1(x — 1) 2 +(.Y - 1) 3 - - 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....

x2 X3 X5 x7( b ,d ) ))1(x ) = 1 —y2(x) _ 1, 2240(a) 3(n +2)an+2 — (n + 1)a„ = 0: n = 0.1.2....

x2 x4 5 6 3 . . (2n — I) ,(b•d) yi(x) = I + 6 +42 1 432 x

6 + 3" 2 . 4 • • • (2,1)2 8 16 2 (2n)

Y2 (X ) = X 1- -X3 X5 +9 135

X7 • •945

• x2"13' • 3 . 5 • • (2,1+ I)

(a) (a + 2)(n + 1)an+2 — (n + 1 )na„_ 1 + (n —1)a„ = 0: n = 0, 1, 2....

X 2 .v3 .v4 x't(11,d) yi(x) = 1 + + + + • • • + + • • • y2(x) =

(a) 2(n + 2)(n + 1)a,.+2 + (a + 3)a„ = 0; = 0,1.2, ...

3(b,d) (x ) = — + — + • • • + (-1)" 3 • 5 • • • (2,z + 1) x2/1 4.

4 32 384 2"(2n)!

x 3 x5 x74 6 • • (211+ 2) ,

Y2 (X) = —+- - ± • • • + (-I)" 2n (2n + 1)!

(a) 2(n • 2)(n + 1)an+2 + 3(tz +1)a„.. 1 + (a +3)a„ = 0; n = 0, 1, 2....

(b) yi (x) = 1 — — 2) 2 + — 2) 3 + X - 2)4 + • • •

y7(x) = (X. - 2) — IOC - 2) 2 + (.v — 2) 3 + 4-(x - 2) .= + • • •

(a) y = 2 + x + X2 -I- X 3 + X 4 + • • - (c) cerca de <0,7

(a) y = —1 + 3x + x2 — - (1, x4 + • • • (c) cerca de lx1 <0.7(a) y 4 — x — 4x2 + Zx 3 + 3.r4 + • • (c) cerca de lx1 < 0,5(a) y = —3 + 2x — ix2 — — + • • • (c) cerca (le Ix' < 0.9(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) A(?. — 4)(A — 8) 621. (a) y1(x) = 1 —

2—

! 4!.4

x2 +61

— 2(X — 2)(X — 6) - 2)(X — 6)(X — 10) 7+ x5 + • • •y2 (x) = x

5! . 3! 7!1, x, 1 — 2x2 , x — 3x3 , 1 — 4x 2 + x 1 ..v — 1,r3 +

1, 2x, 4x2 — 2, 8x 3 — 12x. 16x 4 — 48.v 2 + 12, 32x 5 — 160x3 + 120x22. (b) y = x — .v 3 /6 + • ••

Seciio 5.3

I. 0"(0) = —1, z:/,'"(0) = 0, 014)(0) = 30"(0) = 0, 0"'(0) = —2, (4) (0) = 00"(1) = 0, 0"'(1) = —6, (4) (1) = 420"(0) = 0, 0"'(0) = —a0, 0 14) (0) = —4a1p = p = 00 6. p = 1, p= 3. p = 1

7. p = 1, p = S. p = I

4n (2n — 1)(2n + 1)

RESPOSTAS DOS FROBLEMAS 571

9. (a) p = 00 (b) p = cc (c) p = oo (d) p = co (e) p = 1

(0 p = 4 (g) p = oc (h) p = 1 (i) P = 1 (1) P = 2

( k ) P = 0 ( I ) P = (m) p= oo (n) p= coa 2 (22 a2)a 2 (42 a2)(22 a2)a

2 -10. (a) y i (x) = 1 •rz x42! 4! 6!

6

[(2ni - 2)2 - a2] (22 a2)a2

(2m)!1 - a' 3 (32 - a2 )(1 - a2)

Y2(x) = x 4-- x- +3! 5!

[(2m - 1) 2 - 4 2 1 • • • (1 - a2)(2nz + 1)!

y i (x) ouy2 (x) termina corn x" dependendo se a = tz è par ou imparn = 0, y = 1; n = y = x: tz = 2. y = I - 2x 2: n = 3, y =x - 1X3

11 ' VI (x) = ffix5 "ki x6 + • • • P (x) x kr4 ffi x6TclCi x7 + • • •p = ooy i (x) = 1 - x3 + 1(- 4 . 5 + • • • , y2 (x) = - it,x4 + - 4x6 + • • •p = 00y i (x)= 1 +' + + 45 x 6 + • • • , y2 (x) = x + + kris + 40-x7x + • • • ,p = 7r/2y i (x) = 1 + h.Y3 + 12 x4 - + • • • . y2(X) = X - ix 3 17X5 • • •P= 1Niio é possivel especificar condicöcs iniciais arbitrarias em .v = 0; logo, .v = 0 c urn ponto singular.

x 2 xn- 6.y=l+x+- 4••••+-+ ••=e`2! n!x 2 x4 X6

y= 1 + - + + + + 2 2 • 4 2 • 4 • 6 2" • n!y = +x + + + • • •

119. y = I + x + x2 + • • • + x" +•=

,

xn v3 r'20. y = a„ ( I + x + ,•-7 + • • • + -ni + • • .) + 2 ( v+ :,i-,- .,- • • • +

•2= me + 2( e' - I - x - 2

= cc' - 2 - 2x - x=

Y 2 X4 r6, (- I ),,x2"( 2 22 2! 23 3! 2'n! + • )21. y = ao l - .--- + 2-- - 2- + • • +

.v 2 A.3 X4 .r5

+ (x + 2- - 3- - 2 -4 3 • 5+ + • • •)

= aoc - ' /2 + (x + 2:). - 3x3 - 2 4 + 3 • 5A-5 + . ' ).1. I - 3x 2 , 1 -

\\lOx 2 + 3x4 ; X, x - 3x 3 , X - 13x3 + 4x'

(a) 1, x, (3x2 - 1)/2. (5x3 - 3x)/2. (35x 4 - 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

Secijo 5.4

1. y = c ix I c2x -22. y=c i lx +11 -112 + C2IX + 11-3'23. y = c 1 x2 + c2x 2 In Ix' 4. y = c i x - ' cos(2 In lx1) + c2x - ' sen(2 In lx15. y = c l x + c2x InIx1 6. y = c i (x - 0 43 C2 (X - 1)-4

y = c.214-5-iftzbay = c 1 1X1 312 COS(1 0 In Ix') + c2 1x1 2 sen (1./3' In lx1)y = clx3 + C2X3 In I•Iy = c i (x - 2) -2 cos(2 In Ix - 21) + c 2 (x - 2) -2 sen(21nIx - 21)y = C I IX I -1/2 cos(2 f1.3 In l x 1) + c21x1"" 2 sen ( 1,- 43. In 1x1)y = c l x + c2x413. y= 2x32 - x-'

14. y = 2x- "cos(2 In - x -112 sen (2 In x) 15. y 2x2 - 7x2 In Ix!16. y = x- ' cos(2 In 17. x = 0. regular18. x = 0, regular; x = 1, irregular 19. x = 0. irregular; x = 1, regular

I - xx"n!

I r \ 2 ± 1

1:1l+3) 1/2 )2!(1+:1i)(2-r


20. x = 0, irregular; x ±1, regular 21. x = 1, regular; x = –1, irregular

22. x = 0, regular 23. x = –3, regular24. x = 0. –1, regular; .v = 1, irre gular 25. x 1. regular; x = –2, irregular

26. x = 0, 3, regular 27. x = 1. –2, regular

28. .1 = 0, regular 29. x = 0. irregular30. x = 0, regular 31. x = 0. regular

32. x = 0, ±mr, regular 33. .1 = 0. ±ru , regular34. x = 0, irregular; x ±rur, re gular 35. a < 1

36. fi >0 37. y = 2

a > 1(a) a < 1 e > 0

a < 1eJ3> 0, ou a=lefl>0

a> 1 efl > 0a > e /3 > 0, ou = 1 e > 0a= >0

x-41. y = ao (1 –

2 • 55 + 2 . 4•5 • 944. Porto singular irregular46. Ponto singular regular48. Ponto singular irregular

Se(*) 5.5

45. Porto singular regular47. Porto singular irregular49. Ponto singular irregular

an-2 I. (1) r(2r – 1) = 0: a„ = r1 = r2 = 0(n + r)12(a + r) –11

v .= .v4 X6x ' 2 [1 —

• 2 • 4 • 5 • 9 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

(n + r) 2 –2. (b) r2 – = 0; an = ri = 5 , r2 =

(c) y 1(x) = x 113 [ 1

(-1)"+

m!(1 -4- 3)(2+ 0 • • • (in + 1) (2

r 1 tx \ 2 + 1 (X 4y2 (x) = x -I/3 [1

1!(1 — 1) 1/2 )2!(1 – 0(2 – 1. ) k2

– 5 )(2 – )• • • (In — 1) 1/2. )

t x \ 2,n(–iv,

7)7!(1i

Sugesulo: faca a = 2m na relacilo de recorracia, m = 1.2, 3, ...3. (b) r(r – 1) = 0; a

an_i„ =

; r1 = 1, r2 = 0(a + r)(rz + r – 1)

x x`

(c) Y1(x) = 'x. [ 1 – —1!2! -4- 2—!3(-1)"

! + * + n!(n + 1)!a„_1

4.(b) r2 = 0; a„ = . r i = r, = 0(a ± 62'

x x2

y i (x) = 1 ± + — ± + _ ±aow (2!)2 n(n!)2

C"

5. (b) r(3r – 1) = 0; a„ = – (n + r)[3(tz + r) – 11:

+ •

= r2 = 0

RESPOSTAS DOS PROBLEMAS

(c) Yi(x) = xi" 1 ( x2 ) 2 + • •+1!7 2 2!7 13 ')

(-1)'" x2 m

ni!7 • 13•• • (6m + 1) ( 2) +• "]

(d) y2 (x) = 1 -2

1( 2 ) +115 2!5 • 11 22(x )2 (— 1)"'q"'nz!5 • 11 • (6m - 1)

Sagestiio: faca n = na relacao de recorrencia,m = 1. 2, 3, ...6. (b) r2 - 2 = 0; a„ a- r1 r2 =

(n + r) 2 - 2.v2

y 1 (x) = x12 [1 + +1(1 + 2 V2) 2!(1 + 2 vi )(2 + 2v2)

(-1)" x" + • • -]n!(1 + 24)(2 + 2 i2) • • • (n + 2.4)

x2y2(x) = [1

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-, • • + —n, • = e`

8. (h) 2r2 + r - 1 = (2n +2r - 1)(n r + 1 kl„ +

(-1)"'x'"'ne7 • 11 • • • (4m + 3)( -1)mx2' (d) y2 (x) = x - ' (1 - X2 + X42!5 -.

. .+ 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 = 1

2. v2 2x"

( c) y l (x) = x' (1 + -3 x +— ++ + n!(n + 2) + )(b) r2 - r + 0; (n + r - ) 2a. ± an 2 = = r, = 1/2

r2 _(c) y1 (x) = x 1/2 (1 -—Z +

2• + 2 2 4 22"'(m!)2

(a) r 2 = 0; r 1 = 0, r2 = 0CY(a + 1) a(a + 1)11 • 2 - a(cr + I)](b) y i (x) = 1 + 2 12

(x 1) (2 1 2 )(2 22)

(x 1) 2 + • • •

•

_14_1 y 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

x [1 + E (-1)"(1 + 2a) . • • (2n - 1 + 2a)(1 - 2a) . • • (2n - 1 - 2a) (x 1)"12"(2n + 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)

13. (b) r 2 = 0; r1 0, r2 = 0; a„ = (n - 1 - ).)an-: n2

(-A)(1 - A) 2 (—A)(1 — A.)- • • (n - 1 - A.)(c) y1(x) = 1 + (p (2!)2)2 x + x + +

(n!)2

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,.

573

2a,,_2 = 0:r 1 = r2 = -I

Y2(c) y i (x) = x 112 (17 217 • 11 + • • .)

ti= I

cc1. yi(x)=

E (_1)"x"

n!(n+1)!n=0

Y2(X) = (x) In x + [1 - E x n!(n - 1)! ( 1)".0ii„ + II„


Secäo 5.6(a) x = 0;(a) x = 0;

3. (a) x = 0;(a) .v = I;

(b) r(r - I) = 0; r 1 = 1, r, = 0r2 - 3r +2 = 0; r i 2, r2 = I

(h) r(r - 1) =0; r 1 = 1. r2 = 0(b) r(r + 5) = 0: r 1 = 0, r, = -5

Nab tern ponto singular regular(a) x = 0;(b) T2 +2r - 2 = 0; r 1 = -1+ .7=3 0.732. r, = -I - L=.- -2,73(a ).v = 0: (b)r(r - 0; r: = r2 = 0(a) x = -2; (b) r(r - = 0; r 1 = r2 = 0(a) x = 0: (b) r2 + 1 0: r t = r, r, = -1

S. (a) x = -1;(h) T2 -7r t- 3 --= 0; r 1 = (7 + ./37)/2 -1' 6,54. r2 = (7 - 41)/2 1-1' 0.459

9 (a) x = 1; (b) r2 + r = 0; r 1 = 0, r 2 = -I(a) x = -2: (b) r2 - (5/4)r = 0: r l =5/4, r2 = 0(a) x=2: (b) r 2 -2r = 0; = 2, r2 = 0(a) x = -2; (b) r2 - 2r = 0; r 1 = 2, r, = 0

(a) x = 0; (b) r 2 - (5/3)r = 0; r 1 = 5/3. r2 = 0

(a) x = -3; (b) r2 - (r/.3) - 1 = 0: r 1 = ( 1 + .137)/6 "L- 1,18,r, = (I - .,71)/6 -0,847(b) = 0, r2 =00(c) y1(x) = 1+ x + yx2 + i; •3 + • • •

y, (x) =3 7 1(x) In x — 2x - - + • • .(b) r 1 = 1, r2 = 0(c) y i (x) = x - 4x 2 + - N X4 • • •

Y2(X) -6y, (x) In x + I - 33x 2 + "Tx3 + • • •(h) r 1 = 1, r, =0(c) Y1(X) = -t- + 1X3 + • • •

y2 (x) = 3y, (x) In .v + 1 - 2.4x2 - Vx3 + •(b) r 1 = 1, r2 = 0( c) y i (x) = x ,2 1 4 ,

12' 144"Y2(X) = -Yi (x ) In .v + I - + - +

(b) = 1, r2 = -I(C) Yi(x)=x-1 3+

Y2(x) = (x) In x + - 4.v3 + • • •18. (b) r 1 = r2 = 0

(c) y i (x) = (x -1) 1/2 [1 - i(x -1)+ (x — 1) 2 + - .1. (d) = I19. (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

1 °`' (-1)nx"y i (x) = - Ex n=o (n!)2

(-1)"2"yi = (n!)2

n=0

2N-,-"‘" (-1 ("11„ny2 (.v) = (x ) In x (n!)2n=1

E- (_i)"2"H„Y2(x ) = (x) In x - 2 2

1,1(n!)

14. y i (x)= - E xnx n!(n+1)!n_o

Y2(x) = (x) I n x + 1 11„+11„_1)".e

x2 — 1)!

7. F(s) = s2 b: . s > Ibl

S - a

(s - a) 2 - G2'

s2 + 1)1' s > 0

h13. 12

(s) = (s - a)' + b2'

15. F(s) = (s -

1

a)-, S > a

s' + a-

s > 0

9. F(s) =

11. (s)

17. F(s) =- a) 2 (s+ (1)2'

2a(3s : - )19. F(s) =

(s- + ;

s >

s - a > Ibl

s > a


5. YI(x ) == A3I2 [1 (_ 1.) m \

„,. 1 101 + N2+ 0 • • • (In +

[(:_x• \ 21,1]

Y2(X) = -V-312 1 + E m!(1 - 4)(2 - ) • • • (1,1 - ) 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 continua3. Continua

(a) F(s) = 1/.52 , s(c) F(s) = tz!/sn+1,

F(s) = s gs2 a2),

8. F (s) = s2 -b- b2

s >

10. F(s) = (s - a) 2 - h2

bs a > Ibl

12. F(s) = s , s > 0s2 b2

14. F(s) = S - a

s > a(s - a)2+ b'

16. F(s) = 2as

, , „ s > 0(s- + a- )-

n!

20. F (s) =2a(3s2 a2)

18. I; ( S) =

(s2 - a2)3

on+,

, S > a

> lal

2. Nenhuma das duns4. Seccionalmente continua

(b) F(s) = 2 s3s > 0>0s > 0

s > 0

21. Converge 22. Converge23. Diverge 24. Converge26. (d) 1(3/2) = /2: I 11/2) = 32

Secao 6.2

1. f (t) = ; sen 2t3. f(t) = e t ie-41

5. f (t) = 2e" cos 2t7. f (t) = 2e.' cos t + 3e' sen9. f (t) = -2e-2( cos t + 5e-:' sen t

11. y :(e3r + 4e-2`)13. y = el set) t15. y= 2e' cos t - (2/ ifl)et sen ./517. y = te` - t 2 e: + 313e'

19. .v = cos Nif t21. y= (cos t - 2 sen 4e: cos t -2e`sent)

23. y = 2e-' te" + 212e-1

1 e-` (s + 1)25. Y (s) =

2. f (t) = 2t2e'

4. f (t) = e3' +6. f (t) = 2 cosh 2t - sehn 2t8. f(t) = 3 - 2 sen 2t + 5 cos 2t

10. f (t) = 2e' cos 3t - sen 3t12. y = - e-2:14. y = e:r - te2'16. y = 2e-t cos 2t + ; e- ` sen 2t18. y = cosh t20. y= )w2 - 4) - '[(w2 - 5) cos cot + cos 2t]22. y= i(e-` - e2 cos t + 7e sen t)

S 1 -24. Y(s) - +

s2 + 4 s(s2 + 4)

26. Y(s) = (1 - e-

30. F(s) = 2b(3s2 - b2)/(s2 + b2)3

32. F(s) = n!/ (s - artF(s) = [(s - a) 2 - h'- 1/[(s - a)- + b2]2

a (a + 1)1Y = -1

s 2 (s2 + 1) s2 (s: + 1)29. F(s) = 1/(s - a)2

31. F(s) = rz!/ sn+133. F(s) = 2b(s - a)/[(s - a) : + b2 J 234.36. (a) Y' + s2 Y = s (b) s2 Y" + 2sY' - [s2 +

c)/s2(s2 + 4)

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)

36. EV1 - e-s

= s > 05(1 + e-5)

1 + C"38. r(f (0) =

1 + s2 )(1 - e- T 's > 0


9. (h) f( t) = 1 + 11 2 (0(e- ( ' -2) - 1](b) f (t) = t - u 1 (t) - u2 (t) - 11 3 (t)(t - 2)

(b) f (t) t + 1(2(0(2 - t) + 11 5 (0(5 - t) -

13. F(s) = 2e -s 1 s'e -" e-2"

15. F(s) - -- (1 + 7s)s2

17. F(s) = S

-s22 ((1 - s)e -24 - (1 + s)C3s]

19. f (t) = t3 e2'

21. f (t) = 2u2 (t)e' cos(t - 2)23. f (t) = it i (t)e2(1-1) cosh(t - 1)26. f = 2(2t)"28. f(t) = e'13 (013 - 1)30. F(s) = s- 1 (1 - Cs), s > 0

132. F(s) = - [1 - e S + • • • + e-2"s - e-(2n+11

10. (b) f (t) = r 2 + u2 (t)(1 - 12)

u 7 (t)(7 -

14. F(s) = e -s (s2 + 2)/s3

16. F(s) = 1 (e' + 2e-3s - 6e-4s)

18. F(s) = (1 - e-s)/s220. f (t) = 102 (0[e t - 2 e-2(1-211

22. f (t) 11 2 (r) senh 2(t - 2)24. f (t) tt i (t) + "2 (t) - 11 3 (0 - 114(t)

27. f(t) =t •

cos t29. f(t) = 112(t 12)

31. F(s) = s -1 (1 - e -' + e-2s - s > 01 e-(2n-,21.5

s(1 + e -s ) •s > 0

33. F(s) -.

(-1)"1 / s

s > 01 + cs

n=0

1/s35. 4f - , s > 0

1 + e-s1 - (1 +s)e-`

37. Cif - , s > 0sz(1 _ e-s)

39. (a) ,C{ (0) = s- 1 (1 - e-5 ). s > 0r(g(t)) = S-2 (1 - e'), s > U.C{h(t)} = S-2 (1 - e-') 2, s > 0

- e-540. (b) .4)(0) = s > 0

s2 (1 + e-5)

Sectio 6.4(a) y = 1 - cos t sen t - u3T (t)(1 + cos t)(a) y = e-' sen t + lu,(t)(1 + e - " -') cos t + Sell ti

-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(a) y + le -21 - e-' - t€ 10 (011, - e u-IN(a) y = e-' - e-21 + u 2 (t)(1, - e -u -2 ' + ;e-2"

(a) y = cos t + (t)(1 - cos(t - 37)1(a) y = h(t) - tt, 12 (t)h(t - r/2). 11(t) = (-4 + St + 4e - 't2 cos t - 3co sen 0(a) y sen t + - ite,(0[t - 6 - sen(t - 6)](a) y = h(t) + it, (t)h(t - 7), li(t)= (-4 cos t + sen t + 4e-o cos t + co sen t I

(a) y = 11,(t)[; - cos(21 - )] - 11 3, - cos(2t - )1(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 20/12f (r) = (110)(t - to) - 11 zo+k (1)(t - to - k)1(11/ k)

g(t) = (11 10 (t)(t - to) - 21110 +k (0(1 - to - k) + u,, !-2k ( 1 ) (t - to - 2k)](h/ k)

(b) u(t) = 4ktt3112(t)h(t - - 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

h(t) = a - ICì6 cos( 143 t/6) - (.543/572) e-06 sen(,/iT3 t/6)

19. (b) y = 1 - cos t + 2 E (- o k uk , (t)i 1 - cos(( - k7))k=l

21. (b) y 1 - cos t + E(-1)k uk, (I) (I - cos(t - )1k=1

!I

23. (a) y 1 - cos t + 2 E (-1)k uilk/4(o[ - cos(t - tik/4)]k =1


Secau 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)(a) y = + + 115(t)[-e-20-5, e-u + 1110(0 [1 + e-2(t--10)(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 + u 2,(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)

11. (a) y = 13 cos t + -1sen t - 5-e" itcoscost - 'sen t + , i2 (t)e - ( 4 -N /2 ) sen (t - n/2)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)

t 1 L 2,3613, y ) .1= 0,71153y = (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(a) k l IL 2,8108 (b) k 1 L 2,3995 (c) k 1 = 2(a) 0(t, k) = Ett,s_k (t)h(t - 4 + k) - 114.14(01(t - 4 - k))/2k, h(t) = 1 - cost(b) 0„(t) = u4 (t) sen(t - 4) (c) Sim

20 20

17. (b) y = E u k,(t)sen(t - k7r) IS. (b) y = E(-1)" I lik,(i)sen(1 - k7r)k=1 k=1

20 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 4021. ( 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 = - E20 11 (2k _ (t)e - [4 - (2k - t).,41/20sen{st - (2k - 1)7)/20).1399

Secão 6.6

sen t * sen t = ,; ( scn t - t cos t) e negativo quando t = 27, por exemplo.F(s) = 2/s2 (s 2 + 4) 5. F(s) = 1/(s + 1)(s2 + 1)

6. F(s) = 1/s2 (s - 1) 7. F(s) = s/(s2 + 1)2

9. f(t) = f e -(1-T) cos 2r dr8. f (t) = (t - r) sen r d r

10. f(t) = f (t - r)e-( ` - ` ) sen 2r dr 11. f(t) = f sen([ - r)g(r) drf(In + 1)1(n + 1)

P. (c) I

um (1 - u)" du =1(m + n + 2)

i

13. y= -1

sen (ot + - I senw(t - r)g(r) dr 14. v= e'-`) sen(t - r) sen a r drto co 0 o

y = y e -(`- ` )/2 sen 2(t - r)g(r) drf0

rv = e"12 cos t - le-02 sen t + C(`- ' )/2 sen(t - r)[l. - u, (0] dr

o

f 4v = 2e-2' + 1e-2r + (t - r)e-2('-')g(r) dr

v = 2e-' - e -2 ' + - e -2(' - ' ) ] cos ar dr

1 f'2

19. y = - [senh(t - r) - sen(t - rflg(r) dr' 0

k=1

578 RESPOSTAS DOS FROBLEMAS

f[2 sen(t - r) -sen2(t - r)Jg(r) dr

CAPiTUL 0 7

y = ; cost - 3 cos 2t +

F (s)(13(s) =

1 + K(s)(a) 0(t) = (4 sen 2t - 2 sen t)(a) 0(t) = cos t(b) 0"(0+ 0(0 = 0, 0(0) = 1, 0'(3) = 0

(a) 0(t) = cosh(t)(b) 0"(t) - 0(t) = 0, 0(0) = 1, 0'(0) = 0

(a) 0(t) = (1 - 2t + t2)e-g(b) 0"( t ) + 20' (t) + 0(t) = 0(0) = 1, 0"(0) = -3

(a) 0 (t) = - e l2 cos(,75t/2) + 1-3. e/2 sen(Ot/2)

(b) 0-(t)+0(t)= 0, 0(0) = 0, 0'(0) = 0, 0"(0) = 1

(a) 0(0 = cos t(b) 0( 4)(t)- 0(t) = 0, 0(0) = 1, 0'(0) = 0, 0"(0) = -1, 0"'(0) = 0

28. (a) 0(t) = 1 - .e-1/2sen(0/12)

(b) 0'(t) + 0"(t) + 0'(t) = 0, co) = 1, o'(o) = -1, 0"(0) = 1

Seca() 7.12. = x_, '2 = -2x, - 0,5x 2 + 3 sen t

1. x', = x2 , x'2 = -2x, - 0,5x23. 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)x1 - p(t)x 2 + g(t); x 1 (0) = u0 . x2 (0) = tt'o

7. (a) x1 = + c2 e -3`, x2 = c l e' - c2e-3'c, = 5/2, c2 = -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 - 2x 1 = 0(h) x= 11-e2' - ie-1 , x2 = e2' - le'(c) 0 grafico c assintOtico a reta x, = 2x, no primeiro quadrante.

9. (a) 24 5x', + 2x, = 0x, = - ie/2 _ l e2f , .x2 = e2r

0 grafico a assintOtico a reta x, = x2 no terceiro quadrante.10. (a) xi + 3x1 + 2x, = 0

x 1 = -7e-` + 6e 2r , x2 = -7e'` + 9e'0 grafico se aproxima da origem no terceiro quadrante tangente a reta x, = x2.

11. (a) x',' + 4x, = 0x 1 = 3 cos 2t + 4 sen 2t, x 2 = -3 sen 2t + 4 cos 2t0 grafico 6 urn circulo centrado na origem corn raio 5 percorrido no sentido horario.

12. (a) x',' + x', + 4.25x, = 0x l = -2e- '12 cos 2t + 2e-o sen 2t, x2 = 2e-'12 cos 2t + 2e-ra sen 2t0 grafico a urna espiral se aproximando da origem no sentido horario.

13. LRCI" + LI' + RI = 018. _VI = y3, = Y4, tn iy; = -(k 1 + k2 )yi + k2y2

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) = 15Qi = 42, Qi =36

= qdx, + x2 , x 1 (0) = -17x2 = - 1X2, x2(0) = -21

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 <

+ Fl(t),


Seciio 7.2

6 —6

1. (a) (5 9

2 3

(c) (

6 —12

4 3

9 12

3—2

8

370

(b)

(d)

—15 6 —127 —18

—26 —3

—8 —9 1114 12 —5

5 —8 5

—1—5

2. (a) ( 1 — i —7 + 2i) th \ ( 3 + 4i 6i )—1 +2i 2 + 3i) ' I 11 + 6i 6 — 50

(c)(-3 + 5i

2+i7 + 5i)7+2i

(d) (8 + 7i6 — 4i

4 — 4i— 4 )

—2 1 1 3 —23. (a) ( 1 0

2)—1 (b) 2 —1 1

2 —3 1 3 —1 0

—1 4 0)(c). (d) ( 3 — l 0

5 —4 1

3- 2i 2 — i) (b)

1 — i )

(3 + 2i (c) (3 + 2i4. (a)

1 + i —2 + 3i 2 + i —2 — 3i) ' c ' 1 — i

(10 6 —14(0)5 0 4

4 4

7 —11 —3) 5(0

—16. (a) 11 20 17 (h) 2 7 4

—4 3 —12 —1 1 4)

2 + i

—2 — 3i)

9(c) (

6

—5

8. (a) 4i

( 2s_1110.

—815

—1

(b) 12 — 8i

..._ 1.)

1

—1165

(c) 2 + 2i (d) 16

1

11. ( li

11 —2

1 —3 2 313

P. —3(

3 —1 113.

(

3 — 1

0)

32 —1 0 1_

3 0 3

1 1 12

—s8

14. Singular 15. 0

(

1 _4

)

0 0 12

1 310 10 10

16.. (-110410

1)

to17. Singular

_ 7 1_ 310 10 10

/ 1 1 0 16 y _ 5i

5

18. 0

1

1

1

1

1

))

19.5

0

11-5-

—1—

631

431

0 1 0 1—2 — 5

545

5_ 1

5

5e' 10e2' 2e21 — 2 + 3e3' 1 + 4e-2'21. (a)

(7e—et 7e' 2e21(b)

)4e" — 1 — 3e" 2 + 2e-'

— e' 3e3' + 2e' — e4')+ et 6e' + et + e'r

8e' 0 —e2 r —2e' — 3 + 6e' —1 + 6e.- 2: — 2e! —3e3` + 3e' — 2e4'

et —2e-' 2e2' 1 2e-1(c) 2e' —2e2' (d) (e — 1) 2 — ++ee 11))

—e —3e 4e2' —1 3e -I e +1

580 RESFOSTAS DOS PROBLEMAS

Secao 7.3

1. x 1 = 7X2 = 1 , X3 —

x i = —C, X2 = c 1, X3=

x i = c, x2 = —c, X3 = —c,

5 - X1 = 0, x, = 0, X3 = 0

7. Linearmente independence

9. 2x ( ') —3x( 2) +4x( 3 ) — =0

I I. x") + x(2 ) — x(4) =0

2. Nâo tem solucaoc, onde c e arbitrarioonde c é arbitrario

6. x 1 = c 1 , x2 = c2, X3 = 2c2 + 28. x''' —5x( 2 ) +2x( 3 ) = 0

10. Linearmente independente13. 3x")(t) — 6,C 2) (t) + x (3) (t) = 0

14. Linearmente independente 16. A l = 2, x (1 ' = (13) ; A2 = 4, x (2 ) = (11)

1A2 = 1 — 2i, x (2) = 1 iA l = 1 + 2i, x(1) = (

1 —;1 +( )

A l = —3, x(1) —1)

;., 2 = — 1 , x(2 ' = Ci )A 1 = 0, x (1) = ; A2 = 2, xa)

1)

A l = 2, x" ) = ( A2 = —2 . x121 =( 1 )

•A l = —1/2, x( 1 ) = 3 )(I() ' A2 = —3/2 x(2) = ()

2 0 0;‘, 1 = 1. x( 1 ) =

(

A2 = 1+ 2i, x( 2 ) = 1) : = 1 — 2i, x( 3 ) = (12 i

1 023. A l = 1, x (1) =

( —1; A 2 = 2, x( 2) = 1) ; A3 = 3. x 13 )

0= ( )

1

2 2

124. A 1 = 1, x( 1 ) = -2) :

( -1

A2 = 2, x( 2) = (1) : _ 1, x( 3) 2= (_)

1225. = —1, x (1) = -4 ;

1A2 — 1, X' 2 ' = 0

—1: ;+. 1 = 8, 7e3) = (1)

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 linearmenteindependentes ern todos os intervalos.

Pelo menos um coeficiente tern clue ser descontfnuo em t = 0.

(0 1 x(d) x =—2/-2 2r))

7. (a) W(t) = 1(1 — 2)esx") e x(2) sac) linearmente independentes em todos os pontos, exceto em t = 0 e t = 2; cies silo linear-

mente 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

1 1x = c 1 -4( e -: + c2 0

1 -1

x = c•;

4 e_,, _r (.2 _ .4

-7 __./

1 114. x = c l -4 et + c2 -1

-1 -1

15. x = - 3 (3/ e

4(/1 ) e21 + - (I1)

0 117. x= -2( e' +2 1 e2 '

1 0

20. x =1 11 I t + (3 )t-1

-

122. x = c l (3) + c, t--,4 - 2)

2e -- ` + c3 1 e'l

'?

c_, + c3 (1 e,,

-1

1c -21 + c3 2 ear

1

16. x = 11

18. x = 6

1) e,1

(12

+ ,,1. 0 e3,_ W 1

et + 3 -2 e -1 - 1 eit

-1 1 -8

21. x = (3)12

23. x = c: ( 1

2) (2) t2

)


SecSo 7.52 ?. (a) x = c i ( 1

) e-: + c2 (2) e -2d1. (a) x = c ) (21 )e" + c2 (1)e2t1 3

3. (a) x = (- I CI ) ei + c2 (31 )e' 4. (a) x = c i ( -41 )e-31 ± C2( 1 ) e2t

1 , 15. (a) x = e l (2) e -3' + C2 CI) e-t 6. (a) x = c 1 (_ i)e + c2 ( i ) e-,,

7. (a) x = c 1 (4)

) + c 2 (2) C -2( 8. (a) x =- c 1 (-2 1) ÷ c2(-31)e`

1 . C1 (2 i\ e; _ m i 1\ _„9. x = el()i -i+ c2 ( 1 ) eV 10. X -1 ) c--A-lr

1 111. x - c i (1) e4' + (-2)e`

1 1

1+ c3 ( 0) e'

-1

(a) = x,, .e2 = -(c/a)x 1 - (b/a)xz

(a) x = 2 e - • +55 C) ti.20 29 1 ) e14

(c) 7' 74.39

31. (a) x = c1 e` -2+4)02C2

(4) e1-2-4)//2.1

r1.2= (-2 ± 4)/2: no1

x = c 1 ( -1) e(-1+1211 + e2 (.72) r1.2 = - 1 ± ponto de sela

= -1 ± „AZ, a = 1

32. (a) (v) = c l (31 ) e -2' + C2 ( 11 ) e-t

Sectio 7.6

33. (a) ( 1 4 0

CR2 L ) CL

2t1. (a) x = clef (cos 2tcos

+sen 2t) + czet (- cos 2/ +sen2t

sen 2/

COS —sent10. x = e2t ( cos — 5 Sent

—2 cos t — 3sen t9. x = e' (cos t — 3 sen


2 cos 2t(a) x = cie" ( sen 2t

+ c2e-t —2 sen 2t

cos 2t

5 cos t 5 sen t(a) x = (2 cos t + sent

+ C2 ( — cos t + 2 sent)

5 sen it(a) = cie'12 3(cos 5 cos

it+ sen it)) 2 e

c 112 (3( — COS i t + sen it)

cos t(a) x = cle

)"

sen+ QC' (

— cos t +t

2sen tz cos t +sent

( —2 cos 31 —2 sen 3t(a) x = c i -r

cos 3t + 3 sen3t sen 3t — 3 cos 3t)

0 0x = c l (-3) e t + c2e (cos 2t) + c3 e 1 sen 2t

sen 2t — cos 2t

sen f t fcos t8. x c1 (-2) P -2' cos f t + c3 e sen ./2 t

1 — cos 4 r — ,r2- sen t ../1 cos 4 — sen t

11. (a) r = 12. (a) r = i(a) r a i (b) a = 0(a) r = (a ± — 20)/2 (b) a = 0, ./Y)

15. (a) r = ± 14 — 5a (b) a = 4/5 16. (a) r = 1 ± 107e (b) a = 0, 25/12(a) r = —1+ V:-Te (b) a = —1, 0(a) r = ± 2./49 — 24a (b) a = 2, 49/24

(a) r a — 2 ± Va 2 + 8a — 24 (b) = —4 — 21T), —4 + 211b, 5/2

(a) r = —1 ± 125 + 8a (b) a = —25/8, —3

21. x = cos(41n t)sen(121n t)sen(4 In I)) C2t COS(I In t))

( 5 cos(ln t) 5 sen(ln 0"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,

(25. n 1 = _ _ in ( cos0/2) ) + . _ 02 ( sen(t/2)( ) \/ ) `le 4 sen(t/2) c ' e —4 cos(t/2))

Use c 1 = 2, c2 = — i na resposta do item (b).

lim 1(t) = lim V (t) = 0; naoI-. :"C 1.-,•W

26. (b) (v) = cie-'( cos t sent

+ c,e"— COSI — sen t —sen t + cos t)

Use c1 = 2 e c2 = 3 na resposta do item (b).lira 1(t) = lim V (t) = 0; nä°

(b) r = ±i11717n (d) Irl e a frequencia natural.

(c) ri = —1, ") = 23 ) ; r2 —4, (2) = ( 3 )—4x 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., sen2tx'1 = —3c 1 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

to

(4) = ( — 1)

cos 13 t \— cos 0 I

—0- sen0 t.173 sen13 t /

sen 13 t(— sen 0 t

± C4Nij COS N/73 t

— 0 cos 13 t


0 0 1 0

30. (a) A = ( °403

00

10

9/4 —13/4 0 01 i\

(b) r 1 = i. 11) = ( 1. ) : r2 = —i, (2) =—i

i —i )

4\

r3 = i i, (3)= (

— i t I

r4 = —li ' (4) = —1-03'—3 .10i '15 •

4

2( 15

( cos t \cos t

(c) y = el —sentsent j

(e) c 1 = 10 C2 = 0, C3 = C4 = 0.

0 0

31. (a) A= ( 0—2

1 010 0

0)10(

11 \

(b) = (

i

r2

1—1

r3 = 0i. (3) —A ) ;

(

r4 =

—vi

/ 4 cos P \ / 4 sell; t \

—3 cos P —3 seri i t+c3 +c4

—10sen .. t 10 cos Z t

15 15 5\ 7 sentt / \ — T COS 1 J

period° = 4n.

(2) =

+ C2 (

sentsentCOS I

cost

— 0i,

(c) y = c 1(

cos t \cos t

cos tsent J 4- c'2

/

costtt ± c3t

)—sent

(e) c, = 1. C2 = 0, C3 = —2, c4 = 0.

Sec:10 7.7

=

)1. (b) (I)(t) 3

+ 3 e2' 3e-r _2 e2r

3

32e - I 5e-, 3 3

` — 1 e2t

li e -s/2 + -;e"(

e—t/2 — el2. (b) (NO =

1 e -g/2 — l e- ' le-0 + Ict

4 2 2

(b) OM .----ifr

ef — 1 e-' — 2-— C'(

2 2

—1e` + le'

1e + 2-i 3 - I

(b) ^(t) =le-31 + ie2r 1-

_ + 16,2,5

-3t -1 e2t

5. (b) ^(t) = cos t + 2 seri t —5 sen tsent cos t — 2 sen

( e" cos 2t —2e' sen

cl)(t) = sen 2t2 e' cos 2t)

12

e2' + e e2r 12 ea,

(NO = 22 2 2-+ 2e4( 3e —2: 1 4e-2 e

(b)

(b)

AL_


8. (b) (1)(t)(e-1 cos t + 2e' sen t

5e" sen re'sent

e-t cos t — 2e-`sent

( —2e-2( + 3e - ' —e-21 + e-1 _e-2t -1

e — e + e 5 - lt - ,I-4 -Fe - - 4

r 13—,e —12 e 4- ?: e-D — .4 e' + le

Z e-2' — 2e-' — 2 e2c e — e —2'(b) 4)(r) -= -2i 4-1 i2( ,

7 -2t 2 -t 3 ,t4- )7 , e-

12

1_ ie-21 — ie.- 1 — --2.e212

l e' + l e-2' + 1 e3'6 3 2

^(t) = —der — le- ' + e3'(1 r 1 - 1 r 3t 1 1

— 6 e — -3 e - + -2 e

— 1 e' + 13 e-2t)

ie.' — 11

e-2(

r -2t

-1 e t — e -2t — 1 e31

(b) —2e' + e- 21 +2 e3(

1 t -21 . 1 11— e + e —. -2 e-., e — -3 e

11. x = —71

( I ) e` — 2 (

3) e-t 12. x = ( 3) e- ' cos 2t + c'sen2t

17. (c) x

( /to ) cos wt +vo sen wt

vo —(02110

Seca() 7.8

(c) x = (1) et + c2 [(I) te t + (0) et]

(c) x = ci (,)) + c2 [0) r — (?)]

(c) x= c 1 ( I ) e-r + 2 R I ) + (2) e-]

(c) x = c i ( i ) e-1/2 + c2 [( 11 ) te (1-, + () e

_, ]

0 ,..,

—35. x = c i4 e - ' + c2 e2' + c . re' + 0) e2'

2

• ( 0

—1

( 0

—1

1

1 1 1

1 _(

1 1((1x = c i1 e2 ' + c 2 0)e -1 + c 3 1 e-`

1 —1 --1

(a) x = (3 -I- 4t) e2+ 4t

—29. (a) x = )ert 2 + 2 ( I-1 )

te(12

( —1) ( 024 1 te' + 3 0 e2'+

—6

0)

4 '1 2

(a) x =__ (i) e- '/2 + 71 ( 5) e-71/2

3 1 ° —7

x = ( I ) t + c2 [(1t In t + (0) t]

1 014. x = 1( I ) + c-,[()1 t-- Int ( I ) t-

1 216. (b) ( 1 ,) = — ) e-r'2 + [(_ 2 ) to -112 + (0) Cr'

8. (a) x= 3) e — 6 (11)

10. (a) = 2 + 14 i ) r3

11. (a) x =


( 0

17. (b) xo )(t) = 1 e2'—1

( 0 (1

(c) x(2 )(1)= 1 te2( + 1 e2(

—I 0

( 0 1

2

(d) x (3) (t) = 1 (1 2 /2)0 + 1 te2 ' + ( e2(

—1 03)

0

(1

t +2

e2

)

(e) kli(t)= ' 1 t + 1 (12/2)+r

—1 —t —(12/2)+3

0 1 ') —3 3 2

(f) T= ( 1 1 0, T 1 = 3 —2 —2

—1 3 —1 1 11 0

J= 0 2 1

0 0 2

(1 018. (a) x 111 (t) = 0 e', x(21(t)= 2 et

—3

(d) x(3) (t)= 4_.1

0t et + 0

—1

er

1 0 2: 1 2 2t(e) kl“n = e : 0 2 4t

(ou e' 0 4 4t

2 —3 —2: — 1 2 —2 —2t — 12 0 1 —1/2 0

(f) T= 0

(1

4 0, T -1 = 0 1/4 0.

2 _—2 - 1 1 —3/2 —1

1 0 0

J = (0 1

0 0 1

2x19. (a) J = (2

;k2)0

1

'J3 =

;. 33i.2 (;.4.1 =

0 ;0 0

4;0)A4

exp(Jt) = e'•'0 1 )

x = exp(Jt)x°

1 0 0 1 t2/2

20. (c) exp(Jt) = e:" (0

1t 21. (c) exp(J0 = e' ' ( 0 1 t

0 0 1 0 0 1

Secao 7.9

x = c i ( 1I ) et + c2 (3) e' + ; (I ) teg — 4 (31 )e' + (21 ) t — (?)

x = c 1 ( 13 e'` +c, ( ) e -2r — ( 2/3 ) ( —1

- — n/- 11 Nij C

,

+ 2/0)e_i

x = ci os t + sen t,4

(— 5sentCOS I + 2sen t 12c +

2 tcost — (1)

tsent — 1 cost1

x = c l (2) e -3' + C2 (1) e2I — CO e-2( + (01 ) et

C l/2 cos r13. (a) 11 (1) = 2

4e-Y2 sen it2

)e-`/z sen I t

-4e-02 cos It(b) x = C-f/2

4 4 cos It

sen t


x = ( 12) + C2 [ ( 12) - (I)] - 2 (21 ) In t + (25) t -1 - 1-2

x =+ c (- 21) e_ 5 , + (21) in t 38 (21) t s4 (-20

2

1 17. x = c 1 (

1) e3( + c2 ( -2

) e +4 ( )

1 \ ± 2 (11) ter

8. x = c () et + c2 ( 1 ) e-t + (o)

- -4t - 1 VI - I , 1 ( 2 + 1 e,

x = c i ( 17,) e' + c2 (e

v L 1 j. 2 - A te- +

9 -1 - -./2

x = C15 cost (5/2)/ 5sen t ( 0

+ c' 2 . ) + 1/2) t cost - tsent - (5/2) cos t

()cost +sent - COS t ± scot 1 1

12. x = [-I In(sent) - In(- cost) - 3t + 6.11 ( / cost +sent)2 5 cost

+ G•- In(sent)- lt + c2] ( 5 sent-cost 4- 2scnt

t9. x=c 1 ( 1 )e -r2 +c2U e (± (3) - ( 157 ) + \1/-1

, I;61 _t\ _

2

CAPiTUL 0 8

1 I 4 ,3 3 (i)tInt x c () t + c2 ( 1 ) t -I - (2) 2+ ( 1 ) t - I

1 3 \3)t

x c, (2 ) t 2 + C2 ( 1) t-i + ( -2) t 4 - (2)\ 2 + (2) 10 \ I) /

Seca() 8.1

1 (a) 1,1975; 1.38549; 1,56491: 1,736581,19631; 1,38335; 1,56200: 1,733081,19297; 1,37730; 1,55378; 1,72316

(d) 1.19405; 1,37925; 1,55644; 1,726382. (a) 1,59980; 1.29288; 1,07242; 0,930175

1,61124; 1,31361; 1,10012; 0,9625521,64337; 1,37164; 1,17763; 1,05334

(d) 1,63301; 1,35295; 1,15267; 1,024073. (a) 1,2025; 1,41603; 1,64289; 1,88590

1.20388; 1,41936; 1,64896; 1,895721,20864; 1,43104; 1,67042; 1.93076

(d) 1,20693; 1,42683; 1,66265; 1,918024. (a) 1,10244; 1,21426; 1,33484; 1,46399

1,10365: 1.21656; 1,33817; 1,468321,10720; 1,22333; 1,34797; 1,48110

(d) 1,10603; 1,22110; 1,34473; 1,476885. (a) 0,509239; 0,522187; 0,539023; 0,559936

0,509701; 0,523155; 0.540550; 0,5620890,511127; 0,526155; 0.545306; 0,568822

(d) 0,510645; 0,525138: 0,543690; 0,5665296. (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; -0,7842757. (a) 2,90330; 7,53999; 19,4292; 50,5614

(b) 2,93506; 7,70957; 20,1081; 52,9779


3,03951; 8,28137; 22,4562; 61,54963,00306; 8,07933; 21,6163; 58,4462

8. (a) 0,891830; 1.25225; 2,37818; 4,072570,908902; 1,26872; 2.39336; 4,087990.958565; 1,31786; 2.43924; 4,13474

(d) 0,942261; 1.30153; 2,42389; 4,119089. (a) 3,95713; 5,09853: 6.41548; 7.90174

3,95965: 5,10371; 6,42343; 7,912553,96727; 5,11932; 6,44737; 7,94512

(d) 3,96473; 5,11411; 6,43937: 7,9342410. (a) 1,60729; 2,46830; 3.72167; 5,45963

1.60996: 2.47460: 333356; 5,477741,61792; 2,49356: 3,76940: 5,53223

(d) 1,61528; 2,48723: 3,75742: 5,5140411. (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; 1.4078912. (a) 0,587987; 0.791589; 1.14743: 1,70973

0.589440; 0.795758; 1,15693; 1,729550,593901: 0.808716; 1,18687; 1,79291

(d) 0,592396; 0,804319; 1,17664; 1.771111,595; 2.4636en+1 = [2fpan)-11h 2 , I en+Il [1 + 2 max09 .5 i 10( 1 )1] ,e„+1 = e2,„112. lei I < 0,012, le4 1 < 0,022e„,.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 < 0,045e„ +1 = [7„ + (i „) + q5 3 (inflh 219. e„,_ 1 =119 - 15/4-1/2(7„)02/4en , ' = (1 + + 0(701 1121/12/4e„+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 (d) h < 1/.:,F5()7 0.08e„ +1 = -0"(i„)h2(a) 1,55; 2,34; 3.46; 5.07

1,20; 1,39; 1,57; 1,741,20: 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; 1,729561,19515: 1,38125; 1,55916: 1.729651,19516; 1,38126; 1,55918; 1,72967

2. (a) 1.62283: 1,33460: 1,12820; 0,9954451,62243: 1,33386; 1,12718; 0.9942151.62234; 1,33368; 1,12693; 0,993921

3. (a) 1,20526; 1,42273; 1,65511: 1.905701,20533; 1,42290; 1,65542; 1,906211,20534: 1,42294; 1,65550: 1,90634

4. (a) 1,10483; 1,21882; 1,34146: 1,472631,10484; 1,21884; 1,34147: 1,472621,10484; 1,21884; 1.34147; 1,47262

5. (a) 0,510164; 0,524126: 0,542083: 0,5642510,510168; 0,524135: 0,542100; 0,5642770,510169; 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

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: 1.74992en+ i = (38h 3 /3) exp(47„), le„ 4. 1 1 < 37. 758 8h 3 em 0 < t < 2, le r I < 0,00193389e„+1 = (2h 3 /3) exp(27„), len+1 1 < 4,92604h 3 cm 0 < t < 1, 'e l l < 0,000814269e„+ 1 = (4h3 /3)exp(27„). len+11 < 9,85207h 3 em 0 < t 1, le i l < 0.00162854h--' 0,071 19. /r 0,023

20. h 0,081 21. h .1=-'• 0,11723. 1,19512. 1,38120. 1.55909, 1,72956 24. 1,62268, 1,33435, 1,12789, 0.99513025. 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; 1.89346971,7296802; 1.89346981,7296805; 1.8934711

2. (a) 0,993852; 0,9257640,993846: 0.9257460,993869; 0,925837

3. (a) 1,906382: 2.1795671,906391; 2.1795821,906395; 2.179611

4. (a) 1,4726173; 1.61262151,4726189; 1,61262311,4726199; 1.6126256


5. (a) 0,56428577: 0.590909180,56428581: 0.590909230,56428588: 0.59090952

6. (a) —0,779693: —0.753135—0,779692: —0,753137—0,779680: —0,753089

7. (a) 2,96828; 7,88907: 20.8356; 55,59842.96829; 7,88909: 20.8357: 55.59862,96831; 7.88926; 20.8364; 55.6015

8. (a) 0,9257133: 1.285148: 2.408595: 4,1034950,9257124: 1.285148: 2.408595: 4,1034950,9257248: 1.285158: 2.408594: 4,103493

9. (a) 3,962186; 5.108903: 6.431390: 7.9233853,962186; 5.108903; 6.431390: 7.9233853,962186; 5,108903: 6.431390: 7,923385

10. (a) 1,612622: 2,480909: 3,745479: 5,4958721,612622; 2.480909; 3.745479: 5.4958731,612623: 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: 1,7500000,5909091: 0.8000000: 1.166667: 1,7500000,5909092; 0.8000002: 1,166667: 1,750001

Sec5() 8.5(b) 02 (0 — 0 1 (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 It = 0,25. mas 6 instavel Kara h = 0,3.(d) It = 5/13 0.384615 t.".! suficientemente pequeno.

5. (a) Y = t 6. (a) y = 12

Seciio 8.61. (a) 1,26, 0,76; 1.7714, 1.4824; 2.58991, 2,3703; 3,82374, 3,60413;

5,64246, 5,388851.32493, 0.758933; 1.93679, 1.57919; 2,93414, 2,66099; 4,48318, 4.22639;

6,84236, 6,564521,32489, 0359516; 1.9369, 1.57999; 2,93459, 2,66201; 4,48422, 4.22784;

6,8444, 6,566842. (a) 1,451, 1,232; 2,16 133, 1,65988; 3,29292, 2,55559; 5,16361, 4,7916:

8,54951, 12,04641,51844, 1,28089; 2.37684, 1.87711; 3,85039, 3.44859; 6,6956, 9.50309;

15,0987. 64,074131855. 1,2809: 2.3773, 1,87729; 3,85247, 3,45126; 6,71282, 9,56846;

15,6384, 70,37923. (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


(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, 2,82488Para h = 0.05 e 0,025: x = 10,227,y = -4,9294; estes resultados esttio de acordo corn asoluciio exata ate cinco digitos1,543, 0,0707503; 1,14743, -1,3885

9. 1,99521, -0,662442

CAPITULO 9 Seca° 9.1

(a) r, = -1, = (1, 2) T; r2 = 2, V) = (2, 1) T (b) ponto de sela, instavel(a) r, = 2, r = (1, 3) T; r2 = 4. V) = (1, 1) r(b) no, instavel(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(a) r„ r, = -1 ± i; = (2 f i. 1) r(b) ponto espiral, assintoticamente estavel(a) r,, r, = 44 °,44.2 ) = (2 ± i, 1) r (b) centro, estavel(a) r,, r, = 1 ± 2i; r, 412 ) = (1, 1 (b) ponto espiral, instavel(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(a) r,, r, = ±3i; = (2, -1 ± 30' (b) centro, estavel(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, instavelxo = 1, yo= 1; r, = = -4; ponto do sela, instavelxo = -1, y„= 0; r, = -1, r2 = -3; nO, assintoticamente estavelx„ = -2, y„= r„ r2 = ± ponto espiral, assintoticamente estavelxo = y/S,yo = al r,, r, = IFSi; centro, estavel

17. c2 > 4km. nO, assintoticamente estavel; = 4km . no imprOprio, assintoticamente estavel; c= < 4km, pon-to espiral, assintoticamente estavel

Seca° 9.2x = , y = 2e-2`, y x2/8x = 4e-`, y = 2e2t , y = 32x -2 ; x = 4e-`, y = 0x = 4 cos t, y = 4 sen t, .v 2 + y2 = 16; x = -4 sen t, y = 4 cos t, x2 + y2 = 16

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, assinto-ticamente 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(a, c) (0, 0), ponto de sela, instavel; (A. 0), ponto espiral, assintoticamente estavel; 0), pontoespiral, 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, assintoticamenteestavel; (-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

RLSPOSTAS DOS PROBLEMAS 591

(a, c) (0, 1), ponto de sela, instavel; (1,1), no, assintoticamente estavel; (-2, 4), ponto espiral, instavel(a) 4x2 - y2 = c 18. (a) 4x2 + y2 = c

19. (a) (y - 2x)2 (x + y) = c 20. (a) arctan(y/x) - In y2 = c21. (a) 2x2y - 2xy + y2 = c 22. (a) x2y2 - 3x2y - 2y2 c

23. (a) (y2 /2) - cos x = c 24. (a) x2 + y2 (x4 /12) = c

Seciin 9.3Linear e no linear: ponto de sela, instavelLinear e tido linear: ponto espiral, assintoticamente estavelLinear: centro, estavel; nâo linear: ponto espiral ou centro, indeterminadoLinear: no imprOprio, instavel; nä° linear: no ou ponto espiral, instavel(a, b, c) (0, 0); ti' = -211 + 2v, v' = 4u + 4u; r = I ± ponto de sela, instavel(-2, 2); it' = 4u, v' = 6u + 6v; r = 4, 6; n6, instavel(4, 4); it' = -6ii + 6u, v' = -8u; r = -3 ± ponto espiral, assintoticamente estavel(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(0, 4); u' = v' = (- )u- 3v; r = -3; nO, assintoticamente estavel(-1, 2); u' + r, v' = - 4v; r = (-3 ± /f7)/2; ponto de sela, instavel(a, b, c) (1, 1); u' = -v, v' = 2u - 2v; r = -1 ± i; ponto espiral, assintoticamente estavel(-1,1); u' = -v. c' = -2u - 2v; r = -1 ± ponto de sela, instavel(a, b, c) (0, 0); u' = u, v' = ( ,)v; r = 1. 4-; nO. instavel(0, 2); fi' = = (-4)u - (4) • ; r = -1, no, assintoticamente estavel(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

9. (a, b. c) (0, 0); is' = + 2v, v' = u + 2v; r = (1 ± ponto de sela. instavel(2, 1); is' = + 3u, = -2u; r = (-3 ± j377i)/4; ponto espiral, assintoticamente estavel(2, -2); ii' = -3v, v' = it; r = centro ou ponto espiral, indeterminado(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(a, b, c) (0, 0); = - v, v' = is - u; r = (-3 ± 00/2; ponto espiral, assintoticamente estavel(-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

17. (a, b, c) (2, 2); is' = -4u, v' = + (;,)i.f; r = (7 ± ./273)/4; ponto de sela, instavel(-2, -2); is' = 4v, = (4)" - )v; r = (-1 ± 33)/4; ponto de sela, instal/el

3 .(-2-,2), = -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(c) T 77" quando A -+ 0 (d) A =

(b) tic 4,0025. (b) 4,5130. (a) dx/dt = y, dy/di = -g(x) - c(x)y

0 sistema linear é dx/dt = y, dy/dt = - g' (0)x - c(0)yOs autovalores satisfazem r2 + c(0)r + g'(0) = 0


Secäo 9.4

(b, c) (0, 0); u' = (i37 )u, v' = 2 • ; r = -7;. 2: n6. instavel

(0, 2); u' = (4)0, v' = - 2v; r = -2: ponto de sela. instavel

(3, 0); u' = (4)u - v' = (Dv: r = i; ponto de sela, instavel

(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(0, 4); u' = = - 2r: r = -2: nO, assintoticamente estavel(1,0); u' = - (4)v, v' = (-+)v; r = 4; nO, assintoticamente estavel(1, 1); u' = -u - v' = (-1,)u - (+)v; r = (-3 ± 10)/4: ponto de sela, instavel

3. (b, c) (0, 0); it' = (3)u, v' = 2v; r = 4.2: nO. instavel(0, 2); u' = (4)u, V = (-1)u - 2r: r = -2; nO. assintoticamente estavel

0); u' = (-Du - 3v, = (- 14)1 : : r = -7 : nO. assintoticamente estavel+16-); u' = (-1)u - (1)v , v' = (- 8771 )u - (1-1-10v; r = -1.80475:0,30475; ponto de sela, instavel

4. (b, c) (0, 0); u' = (1 )u, v' = (! )v; r = 2: no, instavel2 4

(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(b, c) (0, 0); u' = u, = (4)v; r = nO. instavel(0, ); tt' = 2

= (2.1,)tt - (1,)v: r = nO, assintoticamente estavel2

(1, 0); it' - v, = r = -1. J.7 : ponto de sela. instavel(b, c) (0, 0); ri' = u, = (4)1 . ; r 1. 4, : n6, instavel

(0,3): u ' = z'' = (M u - (4)1 . : r = -512; ponto de sela, instavel0); tr' = (-.)v, = r = - 1 . 4 : ponto de sela. instavel2); u' = -2u + v, = (4)11 - r = (-5 -1- 0)/2; n6, assintoticamente estavel

(a) Os pontos criticos sac) x = 0,y = 0:x = ,la,, y = 0;x = 0,y = E 0,y 6,1a, quando t oo; osvermelhOes sobrevivem(h) Os mesmos pontos criticos que em (a), mas c,la,, y 0 quando os peixes azulaclossobrevivem(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 c 2 - a2 e � 0: (0, 0), (0, E . ,/a2 ). ( 1 /(7 1 . 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,/a 1 .0) 6 urn ponto de sela: (0. 6 21(7 2 ) e um nO assintoticamen-te estavela ,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 selaa„ = 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 instavela„ = 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 instavela„ = 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 selaa„ = 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


(e) P, é urn no instavel para a > 0. P, 6 urn no assintoticamente estavel para 0 < a < 0,5 e urn ponto desela 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 eurn 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 pontocritico.

a = P3 coincide corn P,.Tambem a = 1.

( 1 - 2.v - y -xJ = -(2a - 1)y 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 esta-veis para a > 1. P, 6 um ponto espiral assintoticamente estavel para 0 < a < 0,5, urn n6 assintoticamenteestavel para 0.5 < a < 1 e urn porno de sela para a > 1.

Seciio 9.5

I. (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(2, 0); u' = - it - v, v' = (4)1 . ; r = ponto de sela. instavel(1, 4); u' = - V = (4)1t: r = ( - 1VT. 10/8; ponto espiral, assintoticamente estavel(b, c) (0, 0); u' = ()1 t. = -v; r .4. -1: ponto de sela. insuivel

0); u' = (4 )11 - ()v. = ()t . ; r = porno de sela, instavel(1, +); u' = -u - (4)r. = ( I.T )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, instavel0); 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(b. c) t = 0, T, 2T....: um maxima //Pith 6 um maxima

t = T/4, 5T/4. ...: dH/cit e urn minimo. P 6 um maximo.

= T/2, 3T/2. ...: H 6 urn minima d PI dt e um minimat = 3 774, 7 774, ...: dfildt 6 urn maxim. P c urn minim°.

(a) .,,/a/„/iiy (b) ,J(d) A raziio das amplitudes da presa e do predador aumenta bem devagar quando o ponto inicial seafasta 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), P 3(3,2 - 6a): Pz se move para a esquerda e P, se move para baixo; eles coincidemem (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 assin-toticamente 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; elescoincidem 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-seurn 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; P 3 (2; 2,4) 6 urn ponto espiral assintotica-mente estavel.


14. (b) A mesma populacao de presas, menos predadoresMais presas, a mesma quantidade de predadoresMais presas, menos predadores

15. (b) A mesma populacao de presas, menos predadoresMais presas, menos predadoresMais presas, menos predadores ainda

16. (b) A mcsma populacao de presas, menos predadoresMais presas, a mesma quantidade de predadoresMais presas, menos predadores

Secan 9.7

1. r = 1,0 = t + to, ciclo limite estavel. 2. r = 1.6 = -t + to , ciclo limite semiestavelr = 1,0 = t + to , ciclo limite estavel; r = 3,0 = t + to , solucao periOdica instavelr = 1, 0 = -t + t0 , soluc5o periOdica instavel: r = 2, 0 = -t + t0 . ciclo limite estavelr = 2n - 1, 0 =1+ to,n = 1, 2, 3, ..., ciclo limite estavelr =2n,0= t + to,n = 1, 2,3, ..., solucdo periOdica instavelr = 2, 0 = -t + to, ciclo limite semiestavelr =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 , solucdoperiOdica semiestavel

9. r = f,e = -t + to, solucrio periOdica instavel(a) = 0,2, T= 6,29;6,29; = 1, T= 6.66; µ= 5. T= 11.6011.60(a) x' = y, y' = -x kty - gy3/3

0 < p < 2, ponto espiral instavel; p � 2, no instavelA "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;= 2, A L. 2,60, T 7,65; p = 5, A 4,36, T L 11,60

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 estavelpara -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° 9.81. (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

(c) -2,6667, e l) = (0, 0,1) T ; A2 -22,8277, (2) (20; -25,6554;0)';11,8277, (3) L' (20;43,6554;0)T

2. (c) A 1 :4' -13,8546; A.2, 0,0939556 ± 10,1945i5. (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 ± 1,5242ic = 1 : P1 = (0,8536; -3,4142;3,4142); = 0,1612; -0,02882 ± 2,0943iP2(0, 1464 ; -0,5858; 0,5858); A = -0,5303;-0,03665 ± 1,1542i

12. (a) P i (1,1954; -4,7817; 4,7817); = 0,1893; -0,02191 ± 2,4007iP2(0,1046; -0,4183;0,4183); A. = -0,9614;0.007964 ± 1,0652i(d) T, L' 5,9

(a,b,c) c 1 "=' 1,243(a) P1 (2,9577; -11,8310;11,8310);P2(0,04226: -0,1690;0,1690); n =(c) T2:4: 11,8

15. (a) P1(3,7668: -15,0673:15,0673);P2(0.03318 : -0,1327; 0,1327); n =(b) T4 L- 23,6

= 0,2273; -0,009796 ± 3,5812i-2,9053:0.09877 ± 0.9969i

A = 0,2324:-0.007814±4.007$i-3,7335:0.1083 ± 0,9941i

RESPOSTAS DOS PRE:I I:SUMAS 595

CAPITULO 1 0 Secao 10.1

1. y = - sen x 2. y = (cot ../27r cos fr + sen

y = 0 para todo L: y = c, sen x se sen L = 0y = - tan L cosx + senx se cos L nao solucâo se cos L = 0

5. Nao tern solucäo 6. y = serhlx + xsen../17r) /2 seni2.7r

7. Nao tern solucao 8. y = c2 sen2x + senx

9. y = c 1 cos 2x + 3cosx 10. y cosx

11. y = - 5 + 3 2

12. y = e(1 — e3 )x -I Inx +Nä° tern solucaon.„ = [ (2n - 1)/2] 2 , y„(x) = sen[(2n - 1)x/21: n 1, 2, 3, ...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(ng In In L): n = 1, 2,3, ...(a) w(r) = G(R2 - r2 )/4p. (c) Q e reduzido a 0,3164 de seu valor original(a) y = k(x4 - 2Lx 3 + L3x)/24

y = k(x4 - 2Lx3 + L2x2)/24y = k(x4 - 4Lx3 + 6L2x2)/24

Seca() 10.21. 7' = 27r/5 2. T= 13. Nao e periklica 4. T 2L5. T = 1 6. Nao e periOdica7. T = 2 8. T = 4

f (x) 2L - x em L <x < 2L: f (x) = -2L - x em -3L <x < -2Lf (x) = - 1 em 1 < x < 2; f (x) = - 8 em 8 < x < 9

11. f = L - x em - L <x < 02L(- 1)" r x

713. (b) f (x) = — E sen 14. (b) f (x) =

n=1 11

15. (b) f (x) = - - + Y.'4 1—, 7 r (2n - 1)2

+[2 cos(2n - 1)x (-1)"Isennxi

I?

7r

,,,I1 4 cc cos(2n - 1)7rx

16. (b) f (x) = - + — E 2 72 (2n - 1)2

•L'

n=1cc(b) f(x) = 3L + ., f 2 L cos[ (2n - 1).Trx/L] : (-1riLsen(n:rx/L)1

4 n_ 1 L (2n - 1) 2 7r 2 nrr

[(b) f (x) =- E - - .r. cos T + sen— sen 2n=1 '

2 nit ( 2 ) 2 n7r] rurx

4 ,--,'". sen[(2n - 1)7x/2]19. (b) f (x) = - _,

2,1 - 17r n=L

2 8 °° (-1)" nrr x(b) f (x) r_-_- -3- + 772 cos 2

n=1

1 12 v-,cc cos[(2n - 1)irx/21 2 "'" (-1)" nrr x(b) f (x) = + 7- .

L2 (2n - 1)2 + -rEt. n sen 2,i=i

1 2 sen[(2n - 1)7rx/L]2 7r 2n - 1

2 D.- on+1

20. (h) f (x) - E sen nnxrr

n=1

n=1

[2 cos nitn271,2

2 - 2 cos nit n- T -2 cos flitsenturxcos n;TX

= 0,5193 quandox ->= 0,5099 quandox ->= 0,5050 quandox

596 RESFOSTAS DOS PROBLEMAS

(b) f (x) = — --; 5 cos ng A' + t [4[1 - (-1)n ] (-1)" -1, im x1 1 1 `.--c ( — 1)n —

n=12 n.--: I n3;73

tur i scn212 + 7r z L-' n2

(b) f (x) = i-3 ± 2_,9 x--,° "̀ [162[(-1)" - 1] 27(-1)" 1 nix 108(-1)"

n4 71- 4 n2,72 cosn 7

+ 54 sen nrrx3 3 3= 1 r=i

26. (b) m = 27

Seca° 10.34 e"" sen(2n - 1);rx

(a) f (x) = - E 21z - 1n=1

°‘‘(a) f(x) = -7 -

[ 2 (-1 r tz.v027r cos(2n - 1)x + seni4 E (2n -

I

L 4L cos[(2n - 1)7rx/L]3. (a) f(x) = - + n27 2 On - 1)2

n=1

2 4 (-1)"+1

3 7r- n4. (a) f (x) = - + E , cos n7 r xn= 1

1 2 (-1) " I

n-

5. (a) f(x) = 2- + - E cos(2n - 1)x

7r 2 - 1

25. (b) = 81

28. f f (t) di pode nä() ser periOdica; pot exemplo. seja f (t) = 1 + cos t

ao(a) f (x) = -- + E(a„ cos frig x + 1) „ sen 117T X);

2n=1

1 2(-1)", 1 -11nn-,

ao = -..:.- . an = , On =..) ti-7r2 1/n7r - 4/,1373,

7r E rl -cost 7T (— 1 )" (a) f(x) = - -

4 + cos frIX sennxI 7r 11 2 n

n= I

n 10; maxlel = 1,6025 cm .v = ±7rit = 20; maxlel = 1,5867 em .v ±7rn = 40; maxlel = 1,5788 em x .+7rNao 6 possivel

a parimpar

1 2 1 - cos tur8. (a) f (x) = 2 +-

72 _, n2cos n7I X

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

f(1 ) =Ir

maxlel = 1 emx ± 1

[6(1 cos an)n 2;72 COS

Mr X

2 n712 cos trr Sen—

tur.v

2

= 1,0606 quando x ->= 1,0304 quandox -> 2= 1,0152 quando X —> 2

2x-,c'c (-1)"Z—,itn= 1

n = 10, 20, 40;Não a possivel

f (x) = -2+

SCIIMTX

n=1n = 10; suplel

= 20; supleln = 40; suplelNäo 6 possivel

I I. (a) f (x) = 6 + E

n=1n = 10; suplel

= 20; supleln = 40; suplelNä° is possivel


'` 1"12. ( a ) f 12 (-= - E ny

) sen turx

n= I

n = 10: = 0,001345 em.v = ±0,9735

n = 20: maxlei = 0,0003534 cm x ±0,9864

= 40: maxlel = 0,00009058 em x = ±0,9931n = 4

�13. y = (co set) nt - nsencot)1 (0)2 _ n 2 ), (02n2

y = (sennt - nt cos n0/2n2 , w2 = n2

14. y = E b„(w sennt - nsenwt)1 w(0)2 n2.,) w 0 1.2, 3, ...11=1

y = b „(tn sen nt - n sen Int) I m (m 2 n 2 ) + h„, (sen nu - nu cos mt)12m 2 ,E aftn

4 rsen(2n - 1)t - 1 semi)/Y = 7r E

1 1 1.2 _ (2n - 1) 2 L 2 - 1 w

n=

1 4 tc cos(2n - 1)71 - cos coty = cos cot + -(1 - cos cot) +

(2n - 1)2, I[0; - (2n - 1)27r21

II

Secüo 10.4

I. Impar

2. Nenhuma das duas3. Impar 4. Par5. Par 6. Nenhuma das duas

1 4 ti 1 - cos(n7r/2) mr xf (x) = 4 + 21. E

n2cos -

211=1

4 ‘---,'" (nrr 12) - sen(tur /2) mr x

f(x) = -,_, sen -7r- n2 2

n= 1

1 2 '" (-1)"-- 1(2tz - 1)7rx(a) f(x) = 2 + 77 E

2n - 1 cos

2n.1

22 nn ) n:r x(a) f (x) = >2 nn ( - cos m

lig 2r + - sen- sen

(a) f(x) = 14 sen(2n - 1)x

18. (a) f(x) = rr 2tt - 1

n= 1

2 ( P 2n7rVT nx19. (a) f (x) >2 cos -

3 + cos -

3- 2 cos n7r sen

3n=1

1 I senburx21. (a) f (x) = L

4L cos1(2,z - 1)7r x1/..420. (a) f(x) _ 2 7r n 2 + 71.2 2_, (2n - 1)2

n=1

2/.. ''''sen(n7rx/L)(a) f (x) = - y

7r 1--, lln=1

7r 1,- ,°c 2 117r 'Ix 4

(cos fir nx

(a) f(x) = -4 + -7r 2_, - sen-2

+ n-2 -2 - 1) cos -

2n=1

(a) f(x) = 2E (- 1 )n sen fix..1

4n2 7r 2 (1 + COS mr) 16(1 -cos n:r)1 nrr x(a) f(x) = E IL ns7r3-2

+ n3 rri

se nn=i

4 16 N--,'"' 1 + 3 cos nn tur x(a) f (x) = 3 + 77. . -, n2

cos -4

(0 n

n.1

n=1

E3 6 - 1 _ cos rur mt. x(b) g(x) = -2 + 772 n2 cos -

3'v=16 %--, 1 tin x

h(x) = L n- sen-3

n=1


1 4 cos(n7r/2) + 2n7r sen (n7r/2) - 4 n7rx(b) g(x) =

4- +

n 2 n 2 COS2

n=1

c* 4 sen 2n7r cos (n7r/2)nrrx

h(x) - E senn7,2 2

n,1

5 12 cos n7r + 4 117T X

(b) g(x) = -12

+ E cos2

n=1

1 °c n2 7r 2 (3 + 5 cos n7r) + 32(1 - cos n7r) n7rx

h(x) = E2 n37r 3

sen 2

n=1

1 6n27r2(2 cos nn - 5) + 324(1 - cos n:r) x30. (b) g(x) = -4 + L n42,4 cos

3n=1

='ICY) E [ 4 cos Tyr + 2 144 cos 1I7T + 180 127 Xsen-

tur n37r3 3

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

1. xX" - xX = 0, T' + AT = 0 2. X" - Ax X = 0, T' + AtT = 03. - X(X' + X) = 0, T' + AT = 0 4. [p(x)XT + XT(x)X = 0, T" + AT = 05. Näo 6 separavel 6. X"+(x+A)X =0, Y" Y = 0

u(x,t) = e -4IX1T2 'sen 27rx - e -2500x2f sen 57rxu(x, = 2e-" 2EI Nen (7rx/2) - e -n2"4 sen7rx + sen 2:-rx

100 1 - cos nz 2 2e

1600 sen u(x, = —r E 40

/11

u(x , t) = 160 tsen(mr/2)

e_„2,20600

SC ntur x

7r-40n=1

100 cos(n7r/4) - cos(3mr/4) , 117 I Xtt(x,t) — E e-n-n-01600 n

IT 40n=1

1 u(x, t) =

80 (— E e-"""1/1("sen

lig

7r)n+1

40n =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 = 40, x = 25,73;t = 100, x = 21,95; t 200, x = 20,31(e) t 524,81

200 cc 1 cos nn 2 2e-n ,- n a t 400 sen

"X10, - E

" 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) awx., - bw, + (c - bt5)w = 0 (b) 8 = c b se b 0X" + 11 2 X = 0, ± (x2 _ tt2 ) y r a 2 x2 T = 0

23. r2 R" + rR' + (x2 r2 _ 42 )R = 0, 9, + i.,29

Seca() 10.6

1. u = 10 + ix3. u = 05. u = 07. it = T(1 + x)/(1 + L)

2. u = 30 - r4

4. u = T6. u = T8. = + L - x)/(1 + L)

70 cos n.7 + 50 _0.86n2,2000 sen

nnx9. (a) u(x , = 3x ri+ 2_, g 20

(d) 160,29 s

=0, T' + a 2 ), 2 T =0

n=1

c„ =

(b) w(x, t) = E c„e-'2#2`146° sen177 X20 '

n=1

3 m4n420[ 3m 3 a 3 (3 cos nur - 1) + 60 cos nut


10. (a) f (x) = 2x, 0 < x < 50; f (x) = 200 - 2x, 50 < x < 100

(b) u(x, t) = 20 -

+ E sen —

cne-1,14n2n2t/(100)2 trr x

5 100800 tur 40

c en = _ s nn 2 a 22 tur

(d) u(50, t) 10 quando t oo; 3754 s

11. (a) 1i(X , = 30 - x + E c„e-n2jT2I/9°3 sen T x30

n.I0

c,, =6

[2(1 - cos na) - n2 a 2 (1 + cos rut)]11 3 7 3

12. (a) u(x,t) = -2

+ ECne-n2 :r 2a2" 2 cos n r XL

n=1

0, n impartc, = -4/(n2 - 1)7 , n par(b) lim,..„„ u(x,t) = 2/n

ti20013. ( a ) / 1 (x , t) = —9

+ Ec„e-n 2 ' 2 "6" cos t40

160cn = (3 + cos na)

3,1272

(c) 200/9

(d) 1543 s25 -'' n2,20900 177 X

(a) 11(X , t) = — + E c„e-' cos 6 30

n=150 rut 1177

Cn = — (sen — - sen —)na 3 6

(2n - 1 urx(b) u(x,t) = Ecne-'2"- n27t2.20L2 sen

2Ln=1

2 Lc„ = L f f (x)sen

(2n - 1)7rx dx

L., 0 2L

16. (a) 11(X , = Ec„e- (211-112.7r2t13W)serl (2n -6

01

)7 x

n=1120

=

, [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.

17. (a) u(x,t) = 40 + E cme.-(2"-1)2'2`/36wsen (2n - 1)ax

40c„ =

(2n - 1)272[6 cos na - (2n - 1)7]

u(x) = I T

L -

a[ + (L la) - 1]0 < x < a,

T[1 a < x < L,a (L1a) - 11

(e) u,(x,t) = sen kt„xa 2 v" + s(x) = 0; v(0) = T1 , v(L) = T2

w, = a2 w„; w(0,t) = 0, w(L, t) = 0, w(x, 0) = f (x) - v(x)(a) v(x) = TI + (T2 - Ti )(x L) + kLx 12 - kx 2 12

160 (cos to - 1)(b) w (x, t) = E c„e -"2 '2114°° sen nnxc„ =

=1 20 n3a3n23. (a) v(x) = T1 + (T2 - TI )x/ L + kLx/6 - kx3I6L

60n=1

onde = K2A2/xIA


Se* 10.7

8 x 1 nit nrrx w rat(a) u(x, t) = „ E -, sen sen -1.,

Lcos .

7. - n=i n

8 x 1 nit 3wr wr x wren(a) u(x,t) = -

n4, E , sen -4 + sen -4 sen L cos L 4

n=1

32 ,--, 2 + cos n7 n7rx rural(a) 10, t) = --i L

!I 3 sen L

cos 7r- L

n=1

4 x sen(n7/2)sen(n7r/L) turx rural(a) u(x,t) = - E sen

rr n Lcos

Ln=t

8L 't•-.., 1 wr tur x nit at(a) Il(X, t) = - 2_, - sen -

2 sen -

L sen-

L(1;7 3 n3n=1

8L N sen(n7/4) +sen(3n7/4) n7rx brat(a) 11(1,0= --=, E sen sen L-cur'

n3 Ln=1

32L x cos nir + 2 sen

tur x nit at(a) tt(x,t) = -4- E

11 .1(ITT

L sen

Ln= I

4L x sen(n7/2)sen(n7r/L) mrx senwr

at(a) tt(x,t) = -7 E sen

TM'

n2 L Ln=1

(2n - 1)7 at0u(x, = E c„ sen

(2n - 1)7rx cos

2L 2L

n=1

Lf

2 (2n - 1)7rxc,,= 7 f(x)sen th

L. 0 2L

8 x 1 (211 - 1)7.(2n - 1)7r (2n - 1)7rx (2n - 1)7r at(a) tt(x.t) = 71,-

2

512 x

n - 1 sen st..n sen cos

4 2L 2 L 21.n=1

(2n - 1)rr at(2n - 1)7 + 3 cos ti7 (2,1 - 1)7rx11. (a) u(x.t) = 74- cos

2L(2n - 1) 4 2 L s,..11

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.

21. r2 R" -1 rR' + (A.2 - ii 2 )R = 0. + IA 2 .0 = 0, T" + A. 2 a = 1- =t) (b) a„ = (1 3 1 + (y 2 L 2 /11 2 7r 2 ) (c) y = 0

20 (2 sen

nit 2tur(a) c„ = sen

n2 7r 2 2 5sen 3'_17

)

Seca() 10.8OG

fir r x I1TT V(a) u(x,y) = E cn sen- senh ' ,

a an=i

4a ,1 sen(n7/2) (b) u(x,y) = -.-., E„,- 4 n2senh(n7b/a)sen

n=1

u(x,y) = E cn sen -senh njr(b - y)117 X

a a n=1

2/a g(x)sen

T X„ =

senh(n7b/a) 10 a dx

2/ Cl X11(X)sen= - dx

senh Our Nu ) fo a

he

IIIT X nary- sena -

a a

,s,3. (a) u(x,y) = Ecn(l) senh sen + E c,,(2) sen senh

n7(1) - y) ,

117T X wry tr. r x

h h a an=1 n=1

- 2/b h wr y 2/(r . FUT X

senh(wr a/ b) f f (y)sen dy, (1,2) =b senh (tur b / a) f h(x)sen

aeh

(b) c 2 2 (n 2 7 2 - 2) cos n:r + 2

l," - c(2) = n7r senh(n7a/6)' "

-IV IT '3 3 senh (nrr b I a)

Co5. it(r, 0) = + E r-"(c„ cos nO + k„ sen 110);

n=tan 2:r

c„ = - f f (0) cos nO dO,7 f

(1)cn

k„ = -f (0)senne dOan f

7 0

RfSPOSTAS DOS PROBL EMAS 601

2 f6. (a) u(r,e) = E en esen ne9 , c„ = —ra" 0

f(0)senne de

u(r,O) = E c„r"'laru

senre

ac, = (21a)a'ia

n. re (B)sen —

re de

tt(x.y) =DO

cn e senE -"Yianrrx2 f" n7 r x

. (a)a

c„ = -a 0

f(x)sen—a

(c) yo -=" 6,6315

2/nir wry10. (h) u(x. y) = Co E c„ cosh mrx cos

wryc„ =

b b senh(nrralb) fo f (

-1 . ) cos —

b dy

n=1

11. u(r,6) = co + E e(c„ cos nO + k„sen ne),n= I

= g(6) cos ne dO, k„ = fo g(0)sen ne de:1nt-r a n I

1

f2D

o2.7

a condicdo necessaria 6 ( g(0) de = 0.

T X n7 r y 2/a aX

(a) li(X y) = E c„sen — cosh —,a

= g(x)sen

cosh(mrb/a) a dxI a4a sen(n:r!2)

(b) c„ =n2 rr cosh(trr b / a)

(2ri - 1prx sen

(2n - 1)rry(a) u(x • y) = E c„stnh

2b 2b

(h) c,, =(2n - 1) 3 7r 3 senh[(2n - 1)7 ral2b]

.coy mix

sen h M

a

MT y,14. (a) li(X, y) = 2— + E ,„ cos

a— senn ,

n=12 a 2/a a n:i X

dxCo = a—

b 0 g(x)dx, c,, -

senh(nrrb/a) fo g(x) cos

a

a4 24a4(1 + cos mr)(b) Co = —

b (1+

30) • C. 's -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 E

1 (2i: - 1) 2 cosh[(2n — 1)7 rb I a]n=

C A PITULO 11 Secio 11.1

1. Homogenea 2. N5o homogenea3. Nao homogenea 4. Homogenea5. Ndo homogenea 6. Homogenea7. (a) 0„(x) = sen n/;C x, onde satisfaz f = - tan ./Tk 7r; (b)

(c) 0,6204, A 2 I" 2,7943(d) A.„ '"=" (2n - 1) 2 /4 para n grande

8. (a) 0„(x) = cos fix, onde satisfaz = cot NA- ; (b) NA°A l -24 0,7402, A2 '="1-.- 11,7349A.„ (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 13,4924X„ (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

n=14 1 cos mr

(b) c„ =Ira" n3

n=1

n=14a2

(b) c„ = (1 - cos nrr)n- 7r 3

n=1

21 b

fb

senh[(2n - )7ra 2h] L f(v)sen (2n 1:ry

2h)

= dy

32h2


k i 111:- -20,1907, A2 2.4 -59,6795A„ z'A -(2n + 1)2 7 2 /4 para n grande

12. ,a(x) =e-`2 13. /2(x) = 1/x14. p(x) = 15. p(x) = (1 - X2)-1/2

X" + XX =0, T" + cT' + (k + Xa 2 )T = 0

(a) s(x) = (b) = n27 2 , 4),(x) = sentur x; n = 1, 2, 3, ..

Os autovalores positivos sao X = onde satisfaz fn = (7)tan(3j.L); as autofuncOes associadassac. Ø„(x) = e-''sen(3,/r;x). Se L = 4, A.0 = 0 6 urn autovalor corn autofuncao associada 0„(x) = xe-2-%

se L #. X = 0 nao é autovalor. Se L < 4, nao existern autovalores negativos; se L > 1/2, existe urn au-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 Cmico autovalor = 0; a autofuncao associada é 0(x) = x — 1.

(a) 2 - cos J.. 0X 1 -1- 18,2738, A2 -1= 57,70752 senhji - cosh ,/Tz = 0, p =

(e) A._ i L, -3,667324. (a) A, = itn, onde p„ e uma raiz de sen AL senhaL = 0, logo A„ = (n7/L)4;

A1 97,409/L4 , X, -25. 1558,5/L 4 , 0„(x) =sen(n7x/L)A.„ = onde pn e uma raiz de senpL cosh AL - cos L senh L = 0;

sen p„xsenh p„L - senp„Lsenh it„xA; -24 237,72/L4 , X 2 .1-- 2496,5/L4 , 4,„ =

senh p„LA, = it 4n , onde p„ é tuna raiz de 1 + cosh it L cos it L = A.1 -1, 12,362/L4,

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)]

Seciin 11.2

1. 0„(x) = 4sen(n - ;)7x; n = 2. Ø„(x) = 4 cos(n — ;)7i . x; n = 1,

00 (x) = 1, 00 (x) = 4 cos fur x; n = 1, 2, ...4 cos -4 x

0„(x) = , onde A.„ satisfaz cos Irk„ - \FAT: = 0(1 +seri' „/—;.,01/2

5. 0„(x) = f e' senn7 r n = 1, 2, ...

an = 44 _

• n =-- 1, 2, ...(2n - 1)272

24a„ = (1 cos[(2n - 1)7/41}; n = 1,2, ...

(2/1 - 1)7

a,, - 2 N/72 s(ennini-)2z1)2(7/2);

n = 1, 2, ...2

Nos Problemas de 10 a 13, a„ = (1 +sen2 fATT,) 1/2 e cos ,A7, -A7,sen,f/:, = 0.

sen X„ 4(2 cos - 1)a,, - ; n = 1,2, ... 11. n„ = ; n = 1,2, ...

./ Ana 11 ?••:(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 - ,/;:i x cos L(c) A. 1 = (27)2 /L2 ; 01(x) = 1 - cos(27x/L)

26. A. 1 = 7 2 /4/.2 ; cb i (x) = 1 - cos(mq2L)

(x) - cos kr„ L + cosh L25. (c) =sen VAT, x. onde A.„ satisfaz cos VX7, L - y,/Z Lsen L = 0

(d) A 1 1,1597/L 2. A2 13,276/L2

2./26. a„ = = 1, 2, . .

(2n - 1)7

- . /72 4c„

=2 (2n - 1)272

[1 e -°1-112 isen( 11 1.)7x,4 \(-1)'14-'

Cn = (2n - 1)272n = 1,2, ...


(a) X" - (v/D)X' +i.X = 0, X(0) = 0, X'(L)= 0; T' +ADT = 0

(e) c(x, t) = E a„e - '. ' bre"l2D senAnx, onde ;t.„ = (v2/4D2);ri = 1

4Dp,2, f r)senp„x dx0

(2LDAn2 + rsen2 it„L)(a) ri, +1;11, = Duxx , u(0,t)= 0, it,(L,1)= 0, tax, 0) -co

(b) 10,0 = E b„e- -- Dè"i2D senit„x, onde ;%.„ = p i:: (v2/4D2);o=1

8c0 ,0 2 p 2" 9(2Dtt„e L12 cos tt„L + ve- ' 1- 12 "sentt „L - 21)11n)I)„ =

(v2 - 402 tt .,1)(2LIDI.q, + vsen 2 L)

Seciio 11.3

1. y = 2 E (-1)"-Isenn7rx

n 2 7 - 2)n7

I cos(2n - 1)7xy = -

4 E4 „=1 [(2n - 1) 2 7 2 - 21(2n - 1)272

(2 cos - 11cos x E sen(n7/2)senn7.vy = 2 E 5. y =8

u= %.„ (;,„ — 2)(1 - sen2 ,FA.7;) (n.27,2_ 2),i2,72

6-9. Para calla problem. a solu45o é

(-1rIsen(n - I)7x2. y = 2 E , 2

[(n - i) 2 7r 2 — 2I(n - k)272

y = " On(X),A n — it

II=

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. NoProblcma 8. o somatOrio comeca cm n = 0.

1 1 1 ( 1a = v =

27- 7 2cos 7x + — -

2- + csen7rX

• N5o tern solucao I?. a arbitrario. y = c cos 7.v + al72

13. a = 0. v = csen7x - (x/27) sen7x 17. u(x) = a -(1) - a)xV (X ) = 1 — t t"

[

12

7A:

u(x.t) = N/5 - 441 + 4c;— + ---,, e - ' 1i's sen —I.- :r- N/2 ,

cos 17.„ x20. u(x.t) = 4 E [ ;.„ _ 1

(e e - '^ ` ) +(1 + seri X,)1/2'

./2- senj„ - cos .fl..;)C„ = • an =

VT.„ (1 ±sen 2 2 An (1 sen2VZ)li2.e satisfaz cos j.; - ‘5.7,seniA.7, = 0.

r t(x,t) = 8 E sen(n7/2)e-" 2

'r21) senturx

717'

c„(e' - e"" -112)2 ' 21 )sen(n - 2)7x

n= I

„= I

u(x,t) = f E - 02,2 _ 1

2 .4(2t: - 1)7 + 4.(-1)"C„ =

(2n - 1)272

E

(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)7xIl(X,t) = .v2 - 2x + 1 +

:r , 2,r-111= 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 x,G(x, ․) = _ s,

s(2 — x)/2,G(x, ․) = x(2 —s)/2,

< S < X

x < s < 1

0< s < xx < s < 1

I cos ssen (1 — x)/ cos 1,G(x, ․) sen (1 — s) cos x/ cos 1,

0 < s < xx < s < 1

s,G(x, ․) = x,

0 < s < x< s < 1

Seciio 11.400 1c„

y = E . J0(N/T1 X), C„ = f 1 f (-410(NA;I X) dx / f 4 (fi.„ x) dx,

=1 An — it f 0

0:7, satisfaz .10 (j) = 0.00

(c) y = — ..1)- +cv „

Jo(4, x);A 4—' An — iin=1

1 1 Ico = 2 f f (x) dx; c„ = f f (410(1Z x) dx I f xfi,(Vr„ x)dx, n = 1.2, ... ;

0 f 0

,/Z satisfaz J,;(1i) = 0.I 1

(d) a„ = f xi k W A 72 x) f(x) dx 1 f x4(47, x) dxf 0

c.c. t , :

An .t C x(e) y = E c'i ikc, ;/7, Al. ,„ = f f (Oh ( 1k, -, x) dx / 1 xf,(. ( ,/— x) (Ix-An — A 0 0n=1

.,7, I I(b) y = cn p2„_,(0, c„ =

0 0f co P2„_, 0,- ) (Ix f Pi„ _ 1 ( x ) dx

Xn — A n--- 1

Seca() 11.5(b) 2) = + 1), (1( , 0) = 0. 0 < < 2

u(0, = u(2, ))) = 0. 0 < < 2

u(r,t) E k„Jo(X„r)senA„at, k,, =. f 1 r.10(Xnr)g(r)dr r4(:.„r) d rA n a 0n=.1

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)n= 1

e X„ satisfaz Jo(X) = 0.

(b) v(r , 0) = coJu(kr) E J„,(kr)(b„,sen me + c„, cos I11 0),

rn=1

1 (27r

ir. I ,n (kc) Jo f (0) sen inO d0; in = 1, 2, ...

1 2,

cm = 7J„,(kc) f0

f (0) cos me d9; in = 0,1, 2, ...1 1

8. c„ = f rf (r)Jo(A,,r) dr I f r4(A„r) drf o00 1 1

10. u(p, ․) = E c„,o n P„(s), onde c„ = f 1 f (arccos s)P„(s)ds / f Ps) ds:

n=0 —I

P„ é o n-esimo polinOmio de Legendre c s= cos 0.

Seca() 11.61. n = 21 2. (a) b„, = (-1)"1+1 i21 nut (c) n = 203. (a) b,,, = 2../2(1 — cos in7)/111 3 7r 3 (c) n 1

(a) fo(x) = 1 (b) (x) = 0(1 — 2x) (c) f2(x) ---, —1 + 6x — 6x2)(d) go(x) = 1, g1(x) = 2x — 1, g2 (x) = 6x 2 — 6x + 1P0(x) = 1, PI(x) = x, P2 (x) = (3x2 — 1)/2, P3(x) = (5x3 — 3x)/2

b,„ =

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