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Respostas dos
Problemas
CAPiTULO 1ec a ( ) LI
1. y

/2 quando t

. y se afasta de 3/2 quando tc3. y se afasta de -3/2 quando t. y - -1/2 quando r- oo5. y se afasta de -1/2 quando r -c. y se afasta de -2 quando to7. y . 3 - y. y' =2 - 3y9. y' = y - 20. y' =3y - 1I I. y = 0 e y =4 sac) solucC ies de equilibrio; v se o valor inicial 6 positivo; y se afasta de 0 se o valorinicial é negativo.y =0 c y = 5 sRo solucOe s de equilibrio; y se afasta de 5 se o valor inicial 6 maior do que 5; yse ovalor inicial é m enor do que 5.
v = 0 6 solucäo de equilibrio; yse o valor inicial c nega tivo; y se afasta de 0 se o valor inicial
positivo.
y = 0 c y =2 siio solucaes de equilibrio:y se afasta de 0 se o valor inicial d negativo; y - 2 se o valor
inicial esta entre 0 e 2;y se afasta de 2 se o va lor inicial d maior do que 2.
(j )

6. (c)

7. (g)

8 .h)

9. (h)

0.e)
(a) dq/dr =300(10 -2 - q10'):q cm g. r em h
(h) q - 0 4 g; nao
dl/ /tit =-kV2 / 3 para algum k.
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 - 250 mg
(a) nu,' =mg - kv 2(bv/mg/k(c) k = 2/49
y d assintOtico a t-3 quando t- co7. yquando t ->00
yc , 0 ou -cc, dependendo do v alor inicial de y
y - cc ou -co, depende ndo do valor inicial de y
yo ou -oo ou y oscila, dependendo do valor inicial de y
y - -oc ou d assintOtico a -,./2t - 1, dependend o do valor inicial de y
y

e então dcixa de existir depois de algum instante ti > 0
y

o 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 a o equilibrio ma is depressaem (b) e (c) do que em (a).2. (a) y =5 + (yo - 5)e ib) y = ( 5 /2 ) +(5/2)Je2'
(c) y =5 + (yo - 5)e21
A solucäo de e quilibrio 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
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q(0) = 5000 g (t ) = 5000e-o3"'
T =300 In(25/6) 428,13 min 7.136 h
(b) q
(d)
556RESPOSTAS DOS PROBLEMS
(a) y = ce -°` + (b/a)
(c) ( i) 0 equilibrio e ma is baixo e 6 aproxim ado m ais rapidam ente. ( ii) 0 equilibrio 6 ma is alto. ( iii)
equi l ibrio permanece o m esmo e é a proximado m ais rapidamente.
(a) y e =b) Y' = aY
(a) yi(t)=
y =cc"' + (b/a)
(a) T =21n 18 -14 5.78 meses
(c) po = 900(1 - e -6 ) 97,8
(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)T4,5 (has
1620 In(4/3) / In 2 672,4 anos
(a) u = T + ( 1 0 - T)e-kr
6,69 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) =10 4 (1 - e- 300I ),t ern h: depois de 1 ano Q 9277.77 g
Q' = -3Q/10 4 . Q(0) = 9277,77
(d) Q(t) =9277,77e -", t cm depois de 1
(e) T - = . 2,60 anos
19. (a) q' =-q/300,
(c) nao
(e) r =250 In(25/6)56,78 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/s1 .e) x = 245 In cosh( t /5) m(a) r 2 .4 0,02828 dia -12 .b) Q(t) =100e-"2828'
(b) kr = In 2
(b) Q(t)-) CV =
ano Q 670,07 g
Seciio 1.3
I . Segunda ordem, l inear
3. Quarta ordem, l inear
5. Segunda ordem , nao l inear
15. r =-2
17. r =2, -3
19.r1, -2
21. Segunda ordem , l inear
23. Quarta ordem, l inear
2. Segunda ordem , nil() l inear
4. Primeira ordem , nao l inear
6. Terceira ordem. l inear
16 . r =±1
18. r =0,1,2
20. r =1.4
22. Segunda ordem , ni l° l inear
24. Segunda ordem , nao l inear
CAPiTULO 2eciio 2.1
(c) y =ce-3 ( + (03) - (1/9) + e -2 ` ;  d assintOtica a / /3 - 1 /9uando t
(c)yce 2 t 3e2'/3; ye quando to
(c) y =ce' + 1 +1 2 e - 72; y

quando t

c
(c) y (c/t) +(3 cos 20/4/ + (3 sen 20/2; y 6 assintOtica a (3 sen 20/2 quando t

0
(c) y = ce 2 ' - 3e'; y -> co ou -co quando t

o
(c) y (c - t cost + se n t)/( 2 :quando t -> cc(c) yce - 1 2 ; yquando to(c) y =(arctan t + c)/(1 4. -2,2) ; yquando to(c) y =ce- 1 / 2 + 3t - 6; y e assintOtica a 3t - 6 quando t -* co(c) y =-te" +ct; yo, 0 , ou -oo quando to(c) y =ce' + sen 2t - 2 cos 2t; y e assintOtica a sen 2t - 2 cos 2t quando to(c) y = ce- ` / 2 + 3t 2 - 12t + 24; y 6 assintOtica a 31 2 - 12t + 24 quando to
13 . y = 3e` + 2(/ - 1)e 2 '4. yt 2 - 1)e-272
15 . y = (3t 4 - 4 1 3 + 6 1 2 + 1)/121 216=(sen 01(2
17 . y =(t + 2) e 2 '8 . y =. - - 2 [( 7 2 / 4 , _4 + )1- co s t +sent]
19 . y =-(1 + 0e - 7e, t A 0 0. y =(1- 1 + 2e - )̀It, t A 0
-* 00
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P ESFOSTAS DOS P ROBLEMS557
(b) y =o s t + s se n t + ( a + D e` i2 ; no =(c) y oscila para a =ao
(b) y = -3e ti 3 + (a + 3)e / 2 ; ao =-3
(c) yoo para a =ao
(b) y = [2 + a(37 + 4)e 2 " 3 - 2e-'/2)/(37 + 4); ao =-2/(37 + 4)
(c) ypara a =n o
(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/4t 2 ; ao = 4/72
(c) y ->

uando t

para a = a()
(b) y=(e ' - e + a sen 1)/sen t; no = (e - 1)/sen 1
(c) ypara a =ao27. (1, y) = (1,364312;0,820082)8. yo = -1 .642876(b) y = 12 + A8 8cos2t + sen 2t - 7 3 - e - f i c t ; y oscila em torno de 12 quando to(c) r =10,065778yo = -5/231. yo = -16/3; y -> -oo quando t -> oo para yo = -16/339. Veja o Problema 2.0. Veja o Problema 4.41. Veja a Problema 6.2. Veja o Problema 12.
Seclio 2.2
1. 3y 2 - 2x 3 = c; y¢0

. 3 y 2 - 21n11 +x 3 1 = c; x A -1.y 5 .1 - - 0
+ cos x = c sey 0 0; tambërn y =0; em toda parte
3 y . + y 2 - X 3 ± x = c ;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 parte
y =sen[ln lx1 + cl se x A 0 e <1;ambêm y =±1
7. y 2 - x 2 + 2(e - e - x ) = c; y + e Y 0 0. 3y + y 3 - x 3  =Cem toda parte(a) v =1/(x 2 - x - 6)c) -2 < x < 3
(a) y =2x2 + 4c) -1 < x < 2
(a )= 12(1 - x)et - 11 1 0c) -1,68 < x < 0,77 aproximadamente
(a) r =2/(1 - 21n0)c) 0 < B <
(a) y = -[2 Im1 + x
2
)
+ 1 L 2(c00 < X < 00
(a) y = [3 - 23 1 +

12

c) lx1 < 415.
(a) y =

.14x 2 - 15

c) x >
(a) y = - i(X 2  +1)/2
c) -cc <.x < 00
(a) y = 5/2 - 3X 3 - ex + 13/4c) -1,4445 < x < 4,6297 aproximadam ente(a) y =- 8e -8e - xc) 1x1 < 2,0794 aproximadamente(a) y = Err - arcsen (3 cos t x)1/3c) Ix - 7/21 < 0.6155(a) y = [; (arcsen1]"c) -1 < < 1
y 3 - 3y2 - x - x 3 + 2 = 0, lx1 < 1
y 3 - 4y - x 3 = -1,X 3 - 11 < 16/3u -1.28 < x < 1,60
y=-1/(x 2 /2 + 2x - 1); x= -2
y =-3/2 + ,/2.r - e + 13/4; x = In 2
25. y =-3/2 + jsen 2x + 1/4; x = 7r/4

6. y

an( r 2 + 2x); x = -
(a) y

eyo > 0;=0 seyo = 0;-eY0 < 0
(1)) T =3,29527
(a) y -+4 quuando t -> oo

b) T = 2,84367
(c) 3,6622 < yo < 4,4042
x =
c
y +
ad -
,
bc
In lay + + k ; 0 0, ay' + b 0
a-(e) ly + 2x 1 3 IY - 2 - r 1 = c1. (b) arctan (y/x) - In lx1 = c32. (b) X 2 - 1 - y 2 - cx 3 = 03. (b) ly - x1 = cly + 3 x 1 5 ; tamba1 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; atnbem y =-x 37. (b) 1x1 3 1 x 2 - 5y2 1 = c
38. (b) clx13 =1y2 2
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558 RESPOSTAS DOS PROBLEMS
Sec5o 2.3
1. t =100In WO min Z-1' 460,5 min. Q(t) =120y[l — e xp(—t/60)]; 120y3. Q0e - 0 .2 (1 — e -02 ) lb 7,42 lb
Q(t) =200 + t — [100(200) 2 /(200 + 1) 2 ] lb,<300; c =121/125 lb/gal;
lira clb/gal
6; . 0 5 o e -riso + 25 _051

-5os +5 ,24 se n t(a) Q(t) =
(c) nivel = 25; amplitude = 25 2501/5002 0,24995
(c )30.41 s
(a) ( ln 2)/ranos

b) 9,90 anos

c) 8,66%
(a) k(e" — 1)/r

b)

3930

c) 9,77%
k = $3086,64/ano;1259,920. (a) 589.034,79b) $102.965,21(a) t35,36 mesesb) $152.698.56(a) 0,00012097 ano- I(bo exp(-0,000120970, t ern anos(c )3.305 anosP =201.977,31 — 1977,31e 0 n 2 n ,< t <t1 ,6745 (semanas)
(a )a- 2,9632; nilob) r = 101n 2,9315
(c) r =6,3805
15. (b),8 36. t =In V /In min L- 6.07 min
(a) n O ) =2000/(1 + 0,048t) ' 1 3(c) r

50.77 s
(a) WO=ce -k ̀+ To + kT I (k cos cot + cosencot)/(k 2 + w2)
9,11°F;

" . :4 ' 3 , 5 1 h
R ./k 2 +w2 ; r = (1/co) arctan (w/k)19. (a) c =k + (P/r) + [co — k — (P1r)]e-"Iv: lim c = k +(Pk)
T =(V In 2)/r: T =(V In 10)/r
Superior. T = 431 anos; Michigan. T = 71.4 anus; Erie. T =6.05 anus; Ontario. T =17,6 anos
20. (a) 50,408 mb) 5,248 s1. (a) 45 .783 mb) 5.129 s(a) 48,562 mb) 5,194 s
(a)76.7 ft/sb)074,5 ftc)5 ft/sd) 256,6 s
(a) duldx =b)(66/25) In 10 mi - 1 :=4. 6.0788 mi
(c) r = 900/(11 In 10) s L- 35,533 s
o

nt 2 gv o

)o v
(a) x,

- In

+ — 

„,

 In 1+—

k 2

g

g
26. (a) v =— (mg/k) + Ivo + (mg/k)I exp( — k t /m )

b) r = — gt: sim
(c) v =0 para t > 0
27. (a) v L =2a 2 g(p —b) e =4ira
3 g(p — p')/3E
(a) 11,58 m /sb) 13,45 mc) k > 0,2394 kg/s(a) v =Ri2g1(R + x)0) 50,6 030. (b) x =ut co s A, y =— gt 2 12+ut sen A + h(d) —16L 2 /(u2 co s t A) + L ta n A +3 > H(e) 063rad<<0,96 radf) u =106.89 ft/s, A = 0 ,7954 rad
31. (a) vtt co s A)e-",=— g/r + (u se n A + g/r)e-"
(b) x =it co s A (1— e-")/r, y =— gt/r+ (it se n 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

. 0 < t < 4
3. 7r/2 < t <37r/2

. —oo < t <— 2
5. —2 < t <2. 1 < t <7. 2t + Sy > 0 0112t + 5y < 0. 1 2 + y 2 <19, 1 — ( 2 + y2 > 0 Ou 1 — t 2 + y2 <0,r 0,y10. Em todo o piano1. y, v A 312. turpara n, ±1, ±2 .... ;e -13. y =±iy) — 4t 2 se Yo A 0; It1 < IYoI/2
y = [(1/yo) — se yo #0; =0 se yo = 0;
o interval° 6 Iti < 1/VT se yo > 0; —oo < t <oo se yo
y =Yo/12ty(+ lseyo

  0;
 =0 se yo = 0;
o interval° 6 —1/2y ) < t <oo se yo 00; —oo < t < cc se yo = 0
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R ESPOSTAS DOS P ROBLEMAS 559
y =± 1 ,I i 1 1 1 ( 1 + t 3 ) + y ii;[1 - exp(-3yZ/2)1" < t < co
y -). 3 se yo > 0: y =0 se yo = 0; y -> -oo se yo < 0
19 . y -4  se yo 9; y -> co se yo > 98 . y - - > -oc se yo < 0; y -> 0 se yo > 0
y .- -oo se yo < y, ';',- ' -0,019; caso contrario y 6 assintOtica a ,5-17
(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) fn5o 6 continua cm ( 2, -1)
1

26. (a) y i  (i) =-:2( t ) = --(s)g(s)dsA U)

0) ,,,
28. y =±151/(2 + 5c t 5 )J1/ 29. y = r/(k + ere')y =±E / (a±ee-2ff )JI/2 I
y = ± p 0)(s)ds + c onde WO =exp(21 sent + 2 T t)to
y =1(1 - - 2 ') para 0 < t < 1; y = -!;(e 2 - 1)e -2' para t > 1
y =CI para 0 < t <1; y =e-''''' para t > 1
Secii() 2.5
y =0 ë. instavel
y =-alb 6 assintoticamente e stiivel.y = 0 6 instavel
y=1 e assintoticam ente estave l, v = c v = 2 sao instaveis
y=0 6 instavel

. y= 0 e assintoticamente estavel
6. y =0 6 assintoticame nte estavel

. (c) y =[y„ (I - y„)kt]/[ I + (1 - y„)kt]
y =1 6 semiestavel
y1 e assintoticamente estavel ,y = 0 e semiestavel .y = 1 c instavel
y =-1 e y =1 sac) assintoticamente e staveis,y = 0 6 instavel
y=0 e a ssintoticamente estavel. y=b = la 2 6 instavel
y=2 6 assintoticamente estavel, y=0 6 scmiestavel.y =-2 6 instavel
y 0 e y=1 silo semiestaveis
(a) r = (1/01n 4; 55.452 anos
(b) T = (1/ r)111[0(1 - a)/(1 - /3)a}: 175.78 anos
(a) y =0 6 instavel,y =K e assintoticamcnte e stavel(b) Convexa para 0 < y <Kle, cOncava para Kle <K17. (a) y =K exp{ [In(yo/K)]e-"I

b) y(2) -1' 0.7153K

7.6 x

g
(c) r

,215 anos
18. (b) (h/a),/ kla7r; sim

c)/a<ra2
19. (b k2/2g(aa)2
(c) Y = E y 2 =KE[1 - (E/r)]d) Y„, =Kr/4 para E =r/2(a) Y 1 2 =K[1 (4h/ rK) j/ 2(a) y=0 6 instavel,y =1 e assintoticamente e stavel( b ) Y = Y of[Yo + (1 - yo)e ' t
(a) y =yoe - 1 3 ' (c ) o exp(-ayo/f3)b) x = .vo exp[-ay 0 (1 -31
(b) z = 1/[v + (1 - u) e 1 3 '1c) 0.0927
(a,b) a =0: v = 0 6 semiestavel.
a> 0: y=f e assintoticame nte estavel e y=- instavel.
(a) a <0:y = 0 6 assintoticamente estavel.

a > 0: y = 0 é instave l; y = jti e y =

it() assintoticamente estaveis.
( a) a<0: y=0 e a ssintoticamente estavel e y=a 6 instavel.
a =0 : y = 0 6 semiestavel.
a> 0: y= 0 C instavel ey=ad assintoticamen te estAvel.
pq[e (q-l"' - 1 ]
28. (a) lim x(t) =min(p, q );(t) -ea(q - P)' -pat(b) lira x(t) = p: x(t)
pat +
L
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560 E S P O S T A S D O S P R O B L E M A S
S e c t io 2 . 6
1 . x 2 + 3x +— 2 y = c
3 . x 3 — x 2 y + 2x + 2y 3 + 3 y = c
5. a x e + 2bxy + c y 2 = k
7. c ' s e t t yy cosx = c ; tambem y = 0
9. e ' Y   c o s 2 x + x 2 — 3 y = c
I 1 . N i io 6 e x a t a
y = [ v3 x 2 ]/2, Ix' < ,123-/ 3y [x — ( 2 4 x 3 + X 2 - 8 x — 1 6 ) 1 / 2 J /4 ,1 5. 1 3 = 3; x 2 y 2 + 2x 3 y = c1 9 . X2 2 I n l y l — Y -2 =c; tamb6m y = 021 . xy 2 — ( y 2 — 2 y + 2 ) e Y = c24. (t) = e x p f R(t) dt, onde r = xy26. i t (x)= ce + 1 +7. it (y) = y; xy + y c o s y — s e t t y = cR(y) e2 Y /y; xe2 — I n I Aambem y = 0u(y ) = seny ; e s e n y + y 2 = c0. ,u(y) = y2; xs 3 x y + y4=c3 1 . p ( x , y ) = xy; x 3 y + 3x 2 + y 3  =
S e c l io 2 . 7
( a ) 1 , 2 ; 1 , 3 9 ; 1 , 5 7 1 ; 1 , 7 4 3 9 (b) 1 . 1 9 7 5 : 1 ,3 8 5 4 9 ; 1 .5 6 4 9 1 ; 1 , 7 3 6 5 8
( c ) 1 . 1 9 6 3 1 ; 1 , 3 8 3 3 5 ; 1 . 5 6 2 0 0 : 1 , 7 3 3 0 8 (d ) 1 , 1 9 5 1 6 ; 1 , 3 8 1 2 7 ; 1 .5 5 9 1 8 ; 1 ,7 2 9 6 8
( a ) 1 . 1 ; 1 , 2 2 ;, 3 6 4 ;, 5 3 6 8 ( b ) 1 . 1 0 5 : 1 , 2 3 2 0 5 : 1 , 3 8 5 7 8 ; 1 , 5 7 1 7 9
( c ) 1 , 1 0 7 7 5 : 1 , 2 3 8 7 3 ; 1 , 3 9 7 9 3 ; 1 , 5 9 1 4 4 (d ) 1 . 1 1 0 7 ;, 2 4 5 9 1 ;, 4 1 1 0 6 ;, 6 1 2 7 7
( a ) 1 , 2 5 ; 1 , 5 4 ; 1 , 8 7 8 ; 2 . 2 7 3 6 ( 1 ) ) 1 2 6 ; 1 , 5 6 4 1 ; 1 , 9 2 1 5 6 : 2 , 3 4 3 5 9
( c ) 1 , 2 6 5 5 1 : 1 . 5 7 7 4 6 ; 1 , 9 4 5 8 6 ; 2 , 3 8 2 8 7 ( d ) 1 , 2 7 1 4 ; 1 , 5 9 1 8 2 : 1 , 9 7 2 1 2 ; 2 , 4 2 5 5 4
( a ) 0 . 3 ; 0 . 5 3 8 5 0 1 ; 0 3 2 4 8 2 1 : 0 , 8 6 6 4 5 8
0 , 2 8 4 8 1 3 ; 0 , 5 1 3 3 3 9 ; 0 .6 9 3 4 5 1 ; 0 , 8 3 1 5 7 1
0 . 2 7 7 9 2 0 ; 0 ,5 0 1 8 1 3 ; 0 . 6 7 8 9 4 9 ; 0 .8 1 5 3 0 2
( d ) 0 , 2 7 1 4 2 8 : 0 , 4 9 0 8 9 7 ; 0 , 6 6 5 1 4 2 ; 0 . 7 9 9 7 2 9
5. Converge para y; M i o e s t a d e f in i d a p a r a y <0. Converge para y : d iv e r g e p a r a y < 0C o n v e r g eC o n v e r g e p a r a ly ( 0 ) 1 < 2 , 3 7 ( a p r o x i m a d a m e n t e ) ; d i v e r g e n o s o u t ro s c a s o s9. Diverge0. Diverge11. (a) 2,30800; 2 , 4 9 0 0 6 ; 2 , 6 0 0 2 3 : 2 ,6 6 7 7 3 ; 2 .7 0 9 3 9 ; 2 . 7 3 5 2 12 , 3 0 1 6 7 ; 2 , 4 8 2 6 3 ; 2 . 5 9 3 5 2 : 2 . 6 6 2 2 7 ; 2 3 0 5 1 9 ; 2 , 7 3 2 0 92 .2 9 8 6 4 ; 2 ,4 7 9 0 3 ; 2 5 9 0 2 4 ; 2 , 6 5 9 5 8 ; 2 , 7 0 3 1 0 : 2 , 7 3 0 5 3( c 1 ) 2 , 2 96 8 6 ; 2 . 4 7 6 9 1 : 2 . 5 8 8 3 0 ; 2 , 6 5 7 9 8 ; 2 . 7 0 1 8 5 ; 2 , 7 2 9 5 91 2 .a ), 7 0 3 0 8 : 3 , 0 6 6 0 5 ; 2 , 4 4 0 3 0 ; 1 .7 7 2 0 4 ; 1 ,3 7 3 4 8 ; 1 . 1 1 9 2 51 , 7 9 5 4 8 ; 3 , 0 6 0 5 1 ; 2 , 4 3 2 9 2 ; 1 , 7 7 8 0 7 ; 1 , 3 7 7 9 5 ; 1 . 1 2 1 9 11 . 8 4 5 7 9 ; 3 , 0 5 7 6 9 ; 2 , 4 2 9 0 5 ; 1 , 7 8 0 7 4 ; 1 , 3 8 0 1 7 ; 1 , 1 2 3 2 8
(d) 1 ,87734; 3 , 0 5 6 0 7 ; 2 , 4 2 6 7 2 ; 1 , 7 8 2 2 4 ; 1 , 3 8 1 5 0 : 1 , 1 2 4 1 1
13. (a )1 , 4 8 8 4 9 ; — 0 , 4 1 2 3 3 9 ; 1 . 0 4 6 8 7 ; 1 .4 3 1 7 6 : 1 , 5 4 4 3 8 ; 1 , 5 1 97 1
— 1 , 4 6 9 0 9 ; — 0 , 2 8 7 8 8 3 ; 1 . 0 5 3 5 1 ; 1 . 4 2 0 0 3 : 1 , 5 3 0 0 0 ; 1 . 5 0 5 4 9
— 1 , 4 5 8 6 5 ; — 0 , 2 1 7 5 4 5 ; 1 . 0 5 7 1 5 ; 1 , 41 4 8 6 ; 1 . 5 2 3 3 4 ; 1 , 4 9 8 7 9
( d ) — 1 , 4 5 2 1 2 ; — 0 , 1 7 3 3 7 6 ; 1 , 0 5 9 4 1 ; 1 , 4 1 1 9 7 ; 1 , 5 1 9 4 9 ; 1 , 4 9 4 9 0
14. (a) 0 ,950517; 0 , 6 8 7 5 5 0 ; 0 , 3 6 9 1 8 8 ; 0 , 1 4 5 9 9 0 ; 0 . 0 4 2 1 4 2 9 ; 0 . 0 0 8 7 2 8 7 7
0 , 9 3 8 2 9 8 ; 0 , 6 7 2 1 4 5 ; 0 , 3 6 2 6 4 0 : 0 . 1 4 7 6 5 9 ; 0 , 0 4 5 4 1 0 0 ; 0 . 0 1 0 4 9 3 1
0 , 9 3 2 2 5 3 ; 0 . 6 6 4 7 7 8 ; 0 , 3 5 9 5 6 7 ; 0 . 1 4 8 4 1 6 ; 0 . 0 4 6 9 5 1 4 ; 0 , 0 1 1 3 7 2 2
(d) 0,928649; 0 , 6 6 0 4 6 3 ; 0 3 5 7 7 8 3 ; 0 , 1 4 8 8 4 8 ; 0 , 0 4 7 8 4 9 2 ; 0 , 0 1 1 8 9 7 8
( a ) — 0 . 1 6 6 1 3 4 ; — 0 , 4 1 0 8 7 2 ; — 0 , 8 0 4 6 6 0 ;4 , 1 5 8 6 7
( b ) — 0 , 1 7 4 6 5 2 ; — 0 , 4 3 4 2 3 8 ; — 0 , 8 8 91 4 0 ; — 3 , 0 9 8 1 0
Uma est imat iva razot ivel para y cm t = 0 , 8 6 e n t r e 5 , 5 e 6 . I \1 5 o e p o s s i v e l o b te r u m a e s t im a t i v a c o n f ia v e l
e m t = 1 c l o s d a d o s e s p e c i f ic a d o s .
Um a es t ima t iva razoave l para y em t = 2 , 5 6 e n t re 1 8 e 1 9 . N a o 6 p o s s i v e l o b t e r u m a e s t im a t i v a
em t = 3 d o s d a d o s e s p e c i f ic a d o s .(b) 2 ,37 < ao < 2 , 3 8 9. (b) 0 .67 < ao < 0 ,68
2. Ni io é exata
4 . x 2 y 2 + 2 x y = c
6. Ni lo 6 exata
8. Nã o 6 exata
1 0 . y In x + 3x 2 — 2 y = c
1 2 .
X
2
+
y 2 =
c
x > 0 .9846
1 6 . b = 1: ez v + = c
20 . e s e n y + 2 y c o s x = c
22. x 2 e x s e n y = c
25. i.t(x) = e 3 . '; 3 x 2 y + y 3 ) e 3i = C
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R E S BOSTAS DOS P ROBL EM S 61
Secáo 2.8
1. dulds =(s + 1) 2u) + 2) 2 ,(0) = 0 . dulds =1 - (u) +3) 3 ,(0) = 0
2 k lk
( a ) 0i(1 =c)(/) = e2 ' - 
k=1
"-1)ktk
k!
a) 0„(t) - Ec )1
k =1
( a ) (t) =

-1 ) k + i ( k+1 /(k

)!2"c)1 ( 1 1 n-c (Mt) =

2t - 4
k=1
tn+i
(a) .0„(t) =t
(n + 1 ) !

,c)im„_, „(t) = t
t2k3k-17. a ) P„(t) = Ek -1. 3 . 5 . • • (2k - 1). (a) c),(t) = =2 . 5 8 ... (3k - 1)t332371 11 ( 1 5(a) o l (t) =3;O2(t) = S + 7 .3(i) = 3. 9 + 3 . 7 9 • 11 + (7 . 9) 2 • 1 5
t4411101 3
(a) 0 1 (0 =2 0) =t -0( )
4" 4 + 4 • 76. 10 4- 64 . 13
-1
11 (a0(0 =t.2( t ) =t - t + 0(t8),
7 t 5

4 / 6
-

I- 007).
5 !

!
3 1 1 6
- 00
6 !

t 234tt°
0 2 (t) = - -
2
- +
6
- -F4- - 0(t7),

t 1t/ 54t6

03(1)=-
2

1 205-(17),

t 2tt 5
04 (0 =-t - -
2 
+ -
8
-
60
+-
15
+ 0(17)
Sec iio 2.9
y„=(- (0,9)"yo; y nquandon -> co=yo/(n + 1):„ -+ 0 quandooy„ =yo,/(n + 2)(n +1)/2; „c, quando ny„° 'e n=4kou n=4k-1:

-yo ,e 1 1 =4k - 2 ou n =4k - 3;y„ nao tem limite quando no
y„ = (0,5)" ( y 0 - 12) + 12; „2 quando n
y,, = (-1)"(0.5)"(yo - 4) + 4; „ -> 4 quando n
7. 7,25%. $2283.63
$258,14
(a ) $804,62

b) $877,57

c) $1028,61
30 anos: $804,62/rnès;

289.663,20 total 0 anos: S899,73/mês;
$215.935,20 total
$103.624,62

3. 9,73%
16. (b) u„co quando no19. (a) 4,7263b) 1,223%,c) 3,5643e) 3,5699Probleinas Variados1. y = (c/x 2 ) + (x3/5)
3. x2 + xy - 3y - y 3 =0
5. x 2 y + xy 2  + = c
7. x 4 + x - y 2 - y 3 = c
9. x2 y + x + y 2=
2. 2y+cosy -x-senx=c
4. 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 t 3 t
03(0 = t -
2 !
+
3 !
+
,t- 3t -7t'
04t - -2 ! -3 ! - t3
12. (a) 0 1 0) = -t - t- -
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562RESPOSTAS DOS PROBLEMAS
23. v

t
= - + C•
25. (x'-/y) + arctan(1y/x) = c
27. (x 2 + y 2  +OCT ' = c
29. arctan(y/x) - In NA +y2 = c
31. x 3 y 2 + xy 3 = -4
(a) y = t + (c - 0 - 1
(c) y =sent + (c co s t -en t) - '
(a) v' - [x(t) + b]v
(b) u = [b f 1 2(t) dt + c]/ µ(t),I) =exp[-(at 2 /2) - bt )
36. v =c 2 + In t7. y =In t +c 2 +1
y = (1/k) In 1(k - t)/(k +1)1+ c 2 se c, =

 0;
 
=(2/k) arctan(t/k) + c2 se
- 2 <

=

c 2 se c, = 0; a mb e mv
y = ±1., (t - 2c1)0 C/2 ; tamhem y = c

tigestrio: /1(v) =

m (ator integrante.
y

 c 2 - o- '
cc i f - In 11 + c l  t I + c 2 se ,-L 0; y =;t 2 + c, se c, =0; tambem y = c
42. y- = c,3. y = sen(/ + c 2 ) =en t + k, cost44. 1.1 3 - 2cl y + c 2 = 2t; tambem y =c5. t + c, =y - 2c1)(Y co1/246. ylnlyl - y + c i y + t =c 2 ; tambern y =c7. e = ( t + c2) 2 + c148. y = 1) 3 1 2 - 19. y =2(1 - 0-2y =3 Int -n( t 2 + I) - 5 arctan t + 2 + ; In 2 +
y=;t2 + 2 .
CAPITULO3Secao3.1
1. y = c i e + c 2 e - 3 '

. y =c l e - ' + c2e-2:
3. v = c i e t/2 + c 2 e - ' 1 3

. y =c l e ` 1 2 + c2e'
5. y =c l + c2 e - 5 '

. y =c l e302 + c 2 e - 3 1 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+ c 2 expl(l - 17 )t]. y = e': y - oo quando t-co
10 . y = ;-e - ' - 1 e - 3; y- 0 quando t- cc1. v = 12e0 - 8e0 2 ; - -cc quando t- 00y =-1 - e- 3 '; - - quando t- ccy =A (13 + 5..ii) expR -5 + ../13)t/21+ A ( 13 - 5../i3) exp[( -5 - i13)t/2]; y - 0 quandoD O
y =33) exp[(-1 +(2/./3) exp[(-1 - N/33)//4]; yuando t- co
y =->co quando t -> oo
y=10+20 +;6,-(r+2)/2y -> -oo quando t ->cc
y" + y' - 6y = 0

8. 2y" + 5y' + 2y = ()
y = e t + e - 1 : minim°=1 em t =1 n 2
y =-' +3e 0 2 ;  maxim° e y =a er n t =1n(9/4), y = Den y = 1 n 9
21. a =-2

24 = - 1
y - 0 para a < 0:y torna-se iliniitado pa ra a > 1
ypara a < 1: nao existe a tal que todas as solucOes nao nulas se tornam ilimitadas(a) y = 1 -   (I + 21:3)e - 2 t + 1(4 - 2/3)e(i2.
(b),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,
 „, = + 0) 3 /(4 )4)2
p = 6(1 +- 16,3923d) t„, - In(3/2), y,,,- oo
(a)y =d/cb) aY" +bY ' + cY 0
(a) b > 0 e 0 < c < b 2 /4ab)c < 0 (c) b < 0 c 0 <c < b2/4a
11. (x3 /3) + xy + = c
13 . y = tan(.r + x 2 + c )
15 . y =c/cosh2(x/2)
17 . y = ce 3 x - e2"
19. 2xy + xy 3 - =
21. 2xy 2 + 3x 2 y - 4x + y3 =c
e 2 t
12 . y =ce' + e - `14. x 2 + 2xy + 2y 2 = 3416 . e -x cosy + e 2 Y se n  cx
18 . y =e -2 "  f e
,
cis +
20. e + CY =C
22. y 3 + 3y - x 3 + 3x = 2
24. sen ysen2 x = c
26. CY R + In 1x1c
28. x 3 + x 2 y = c
30. (y 2 /x 3 ) + (y/x 2 ) = c
1
32. - = -x f
S
ds +2
Y
(b) y =r' + 2t(c - (2)-I
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RESPOSTAS DOS PROBLEMAS 563
Seca() 3.2
1 .. 13. e -4 (. x2ex
5. -e 2 1. 0
7. 0 < t < oc. -oc < t <
9. 0 < t <40. 0 < t <oc
11. 0 < x < 3

2. 2 < x < 3:r/2
14. A equacao é nao linear.

5. A equacao é nao homogénea.
16. Nao

7. 31 e 2 : +ce2'
18 . te` + ct

9. 51 ,1/( f ,g)
-4(t cost - sent)
v 3 e y, formam urn conjunto fundam ental de solucties se e somente se a,b 2 - a,b,  .
-2y i (t) =e'2 (t) = + jetv1(t) = -1e - 3 1 1 - 1 ) +2(t) = - .1r e- 3 " -1) ' 11 1 - 1 )24. Sim5. Sim
26. Sim7. Sim
(b) Sim. (c) iy,(t).y.,(t)] e [y,(t),y,(t)] sac) con juntos fundamentais de solucOes; ry 2 (t),y 3 (1)] e b74(t),y,( t)]
nao sao
ct 2 e'0. c cost
31. c/x

2. c/(1 - x2)
34. 2/25

5. 3

= - 4 . 9 4 6
36. p(t) =0 para todo t
40. Se t„ for urn porno de inllexao e se y =5(t) for tuna solucao, entao. da equa cao di ferencial, p(t,,)(p' (t o ) +
(1(t())0((„) = 0.
Sim. y =cie-•212I

'2 dt + c2e 0/2
xu
Nao
'I),os x
Sim. y = I
L
c;f.1-atmcl t (X) = exp [- j (-
v
+ -) ad
12(x)o-
Sim. y =c l x - 1 + c 2 x7. .r 2 u" + 3.r E t ' + ( 1 +x 2 - v 2 ) 1 . = 0
48. (I - x 2 )p" - 2x/1' + a(a + 1)/./ = 09. /1" - xu. = 0
51. As equacOe s de Legendre e de Airy sao autoadjuntas.
Sec:10 3.3
1. e cos 2 + ie sen 21,1312 + 2,4717i
. e2 cos 3 - ie 2 sen 3 -.4_ -7,3151 - 1,0427i
-1
e 2 cos( g /2) - ie 2 sen( g/2)e i7,389112 cos(In 2) - 2isen (In 2),5385 - 1,27791
r cos(2In>r)+ i;r - t sen 2 In ir)0.20957 + 0.239591
7.= cle t cos t + c 2 e`sent. y =c l e f  co sc 2 esen9. y = c l e 2 1 +c 2 C'0. y =c l e' co s t + c2 e - i sen t
11 . y =c l  e- 3 1 cos 2t + c 2 e - 3 r sen 2t2. y =c 1  cos(3t/2) + c2sen(3t/2)
13 . y =os(t/2) + c 2 e -' son (02)4. y =c i e ( 1 3 + c2e-403
15 . y =c l e - 1 1 2 co s t + c2ea sent6. y =os(3t/2) + c2e-2rsen(3t/2)
17 . y =en 2t; oscilacáo regular
y = e 2 r cos e - 2 1 se n t;oscilacao decrescente
y =-e'-' / 2 sen 2t; oscilacao crescente
y =(1 + 20) cos t - (2 - 0) sen t; scilacao regular
y =3e-' 1 2  co s t + ;e -''2 2 sen t ;scilacao decrescente
y =../e-(1-7`14) co s t +- ( 1-7/4 'sen t;scilacao decrescente(a) u =2e 1 / 6 cos(if t/6) - (2/03)e 0 6 se n ( fn t/6)(b) t = 10,7598
24. (a) u =2e - 1 / 5 cos(/34 t/5) + (7/ .0-1)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' co s t + ae-a i sen tb) T = 1,8763
(c) a = a, T =7,4284;= 2, T =4,3003; =2, T =1,5116
L._
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564RESPOSTAS DOS PROBLEMAS
36. y = c 1 t - 1 + c 2 t - 2
38. y = c i t 6 + c,t-1
40. y = c l t cos(2 In t) + c 2 1 sen (2 In t )
42. y = c i t - 3   cos(In t) + c 2 t -3 sen (In t )
35. y = c 1  cos(In t) + c2sen(Int)
37. y = c 1 t - 1  cos(; In t ) + o t - 1  sen (; In t )
39. y = c i t 2 + c,t3
41. y = c i t + c 2 t -3
Sim, y = c i cos x + c 2 sell x, x= f e- ' 2 1 2 d r
Niio
46. Sim. y = c l e - ' 2 1 4 cos(ijt 2 i4) + c 2 e - ' 2 1 4 se n ( i 3 t2/4)
Seciio 3.4
2. y =

c 2 te - ' 3. y = c l e + c2te
4. y = c i e -3" + c 2 te -3". r = c i e - ` 1 2 + c 2 e 3 ' I 2
6. y = c l e 3 ' + c 2 te 3 '.y i  e cos 3t + c 2 esen 3t
7. y = c i e -0 + c 2 e - 1 1. y = c l e -3r;4 c 2 t e - 3 ' ' '10. y = Cl/2 cos( t /2) + c 2 e -t i2 sen (t/2). y = c l e 2 ' 1 5 + c 2 t e 2 t /5y = 2e20 - 3l e 2 r / 3 , co quando toy = 2te 3 1 , yo quando to
y = -e - 1 1 3 co s 3t + 9e - t i 3 s e n 3t, y -› 0 quando t - - > 0 0
y = 7e -2 0 ' 1 ' + 5 t e - 2 ( '+ ' ) ,quando t -› co
15. (a) y = e -3 ' I 2 - e - 3 ( / 2b) t = -25to = 16/15,o =1- -0,33649y =b + 4)te - 3 0 2 ;= -;16 . y = 2e 0 2 + (b - 1)te l 2 -; b=17. (a) y = e - ' 1 2 + i te- ' I 2b) t it = 5, ym = 5e- 3 . 5.24664y = P - 4 I 2b + 1)te-t/21M! = 4b/(1 + 2b) -4 . 2 quando b -+ co; y m = (1 + 2b) exp[ -2b/(1 + 2b)] 00quando b -> co18. ( a ) Y = ae - 2 1 / 3 -F (ia - 1)te-2'1323. y 2 (t) = t 325. y 2 (t) = t- I I n r
27. y 2 (x ) = cosx2
29. y 2 (x) = x' 'e-2`ii
Y
32. y = c l e - ' , 2: '

a s e / 2 d s + c 2 e -ix2/2
o
34. y 2 (t ) = t- 1  In t
36. y 2 (x) = x
39 . (b) Y o + ( a / b ) ) / 0
42. y = c i t - 1 / 2 + c 2 t - 1 1 2 In t
44. y = c i  t - 1 + c 2 t - - 1  In t
46. V = c, t - 2 cos(3 In t) + c 2 t -2  sen (3 In I)
( b ) a =
24. y 2 ( t) = t • - 2
26. y 2 (t ) = re '
28. y 2 (x) = x
30. y 2 (x) = x -1/2 c o s . v
33. y 2 ( t) = y i ( t) f yi-2 (s) exp [-f (r) d rids
( 0
35. y 2 (t ) =co s t -
37. y 2 (x) = x - 1 1 2 cos x
41. y = c , t 2 + c 2 t 2 In t
43. y = c i t + C 4 5 1 2
45. y = c, 1 3 /22 t 3 /2 In r
Secao 3.5
y = c l e3 ( +c2 e - ' - e2 r
y = c l e - ' co s 2t + c 2 e -t sen 2t +sen 2t - 11 cos 2t
3. y = c i e 3 ' + c 2 e - ' +. y = c 1 + c,e - 2 ' + t - sen 2t - cos 2t
y = c i  cos 3t + c 2 sen 3t +

9t 2 - 6t + 1) e 3 ` +
y =

c2 te - ' + t 2 e - '
y = c i e - 1 + c 2 e - ' / 2 +1 2 - 6t + 14- sen t - cos t
y c c o s t + c 2  sen t - It cos 2t -

en 2t
u = c l cos wot + c 2 se n coo t + (4 - ( 0 2 ) -1 c o s C o t
u = c i co s w o t + c 2   se n c o o t + (1/2w 0 )t sen c o o t
y =os(il3 t/2) + c,e ' I 2 s e n (../T3 t/2) +b e' -y =Ce-1c . 2 e 2 t3. y =t - l e -2 r-44. y' cos 2t + 1 4 2- C5. y = 4 te - 3e + r 3 e i + 416 . y = re- '7. y = - 2 cos 2t - 18 - sen 2t - it cos 2ty = e - ' co s 2t +en 2t + te-' sen 2t
(a) Y (t) = t(A 0 t 4 + A IP A2I2 4 1 3 1 - 1 - A 4 ) + t ( B 0 t2 -1- Bit2 ) 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
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RESPOSTAS DOS PROBLEMAS 565
(a) Y (t ) = Aot + A 1 + t (Bot + B O s e n t + r(Dor + D i ) c o s r
(b) Ao = 1, A l = 0 , Bo = 0 , B 1 = 1 / 4 , D o = -1 / 4 , D 1 = 0
(a) Y (t) = e l ( A c o s 2 t + B sen 2t) + (Dot + D I ) e 2  s e n I + (Eot + E l ) e 2 ' c o s t
(b) A = - 1 /20 . B = -3 /20, Do = -3 /2 , D I = -5 , Eo = 3 /2 , E 1 = 1/2
(a) Y (t) = Ae - ' + t(Bot 2  + B 1 t + B 2 ) e - ' cost + t(Dot 2  +D I t +D 2 ) e - ' s e n t
(b) A = 3, Bo = -2/3 , B 1 = 0 , B2 = 1 , D o = 0 , D i = 1 , D2 = 1
(a) Y( t ) = Ao t 2 + A l t + A 2 + t 2 (Bot + B 1 ) e 2 i + (Do t + D 1 ) s e n 2 t + (Eot + E I ) c o s 2 t
(b) Ao = 1 /2, A l = 1 , A2 = 3/4 , B o = 2/3 . B 1 = 0 , Do = 0 , D I = - 1 /1 6 .
E 0 = 1 /8 , E 1 = 1 / 1 6
(a) Y(t) = t (Ao t 2 + A t + A 2 ) s e n 2 t + t(B 0 t 2 + B 1 t + B 2 ) c o s 2 t
(b) Ao = 0, A 1 = 13 /16 , A2 = 714 , Bo = -1 /12, B I = 0 , B2 = 1 3/32
(a) Y (t) = (Ao r 2 + A l t + A 2 )e sen 2t + (8 0 1 2 + B i t + B 2 ) e t   c o s 2 t +
e - ' ( D c o s t + E s e n t) + Fe
(b) Ao = 1/52, A l = 10/169, A2 = -1233/35.152, B o = -5 /52 . 8: = 7 3 / 6 7 6 ,
B2 = -410 5/35.152. D = -3/2, E = 3/2, F = 2/3
(a ) Y( t ) = t (Ao t - AO C ' cos 2 t + t ( B o t + B i ) e - ' s e n 2 t + ( D o t + D ; ) e - 2 r c o s t +
(Eot + E 1 )e- 2 ' s e n r
( b ) A o = 0 , A l = 3/16, Bo = 3/8 , B 1 = 0 , D o = -2/5, D 1 = -7 /25 . E 0 = 1/5 ,
E l = 1 / 2 5
(b) to = (- le
28 . y = c 1  c o s At + c, s e n A t + E ta „,/( A 2 — m 2 7 2 ) 1   s e n m a r t
rn.I
I. < t < 72 9 " Y = 1 -( 1 + 7 / 2 ) s e n t - ( 7 / 2 ) c o s t + (7/2)e', > 730 . y = I- e - • - : sen 2t - 1e - ' c o s 2 t ,< t < 7 /2S e c a ° 3 . 6
1 . Y ( t) =. Y ( r ) =
3 . Y ( r ) = ,1 r 2 e - g. Y( t ) = 2 r 2 e 1 2
y = C I c o s t + c 2  sen t - (cost)) In(tan t + s ect)y = c i c o s 3 t + c 2 s e n 3 t + ( s e n 3 t ) In ( t a n 3 t + s e c 3 t ) —
7 . y = c

e- 2 ' I n t
y = c 1 c o s 2 t c 2  sen 2t + (s en2r) In sen 2t - it cos 2t
y = c i cos(t/2) + c : sen (t/2) + t sen (02) + 2[In cos(t/2) I cos(t/2)
10. y =c 2 te -t In(1 + t 2 ) +r c t a n r=c l e 2 ' + c,e 3 ' + f [e 3 1 - s ) - e 2 ( ` - s ) ] g ( s ) d sY = c 1 c o s 2 t + c 2   sen 2 t + f s e n 2 ( t - s ) 1 g ( s ) d s1 3 . Y(t ) =; + f 2 I n t4 . Y ( t) = -2/ 2
1 5. Y(t ) =1( t - 1 ) e 2 t16(t ) =-2(2t - 1)e-1
17 . Y( . v ) = 1 . x 2 ( I n . 0 318 Yx) =- ix1 2 c o s x
19 . Y (x ) = f (A :1 e t- r ) -ff i g(r) e ll

0 . Y(x) = X - 1 / 2 f t - 3 / 2 sen(x - t)g(t) di
(
(b) y= yo c o s t + y ' , s e n t +e n (t - s)g(s)dst ofi(y = (6 - a) I f [e'''' ) - ea('-')]g(s) ds5. y = it - 1 1 e ( ` - ' 5 ) s e n µ(t - s)g(s)dst oiii 29. y = c i t + c 2 t 2 + 4 1 2 I n t6 . y=f t - s)e a u ' 3 g ( s ) d sy = c 1 t - I + C 2r 5 + i 4 i1 . y = c 1 (1 + t ) + c 2 e + 1 (t - 1 )032. y = c 1 e ` + c 2 1 - ;(2r - 1 )e-`-1(1 +  ) e - ' cos 2 t - ( 1 + e ` 1 2 ) e - '   s e n 2 t ,> 7 /2Ni io4 . yc 2 e - ` -1 1 .1 2 .
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566 RESPOSTAS D OS PRO BLEMAS
Seca() 3.7
I t = 5 cos(2t - 6).= arctan(4/3),9273
u =2 cos(t - 27r/3)
u =2./3 cos(3t - S), S =-aretan(1/2)0,4636
uif f cos(7rt - 6). S =n + arctan(3/2),1244
uos 8t ft, t ern s:8 rad/s,=7r/4 s, R =1/ 4 ft
u

 sen 141 cm. t ems:=7r/14 s
u =(1/44) sen (841) -

os(84 t) ft, t em s; w =8f rad/s,
T = 7144 s, R =V11 288 L'" 0.1954 ft,= 7r - arctan(3/Nif) 2.0113
Q = 10 - 6 cos 20001 C, t em s
u = e -10'[2 cos(lig t) + (51.A) sen(4.A 0 1 cm, tem s;
tt =4f rad/s.T,=r2"6 s,T,/T =7/2.A -='" 1,4289, r :1=_, 0,4045 s
u1/8,/31)e - 2 : sen (2) ft. terns;= 7r . /2./31 s" :=-=1.5927 sttl' 0,057198e -m 5 ' cos(3.87008 t - 0,50709) m, t ern s;3.87008 rad/s,ttlak, = 3,87008/ ./1 7 - 0.99925
Q = 10- 6 (2e - 5 m r - 6.- 1 ° ° c ') C; t em s
13. y= ..4907
r =,M + B 2 , r cos 6 = B. rsen 9 = -A;=r; S =0 +(4tt + 1)7/2,
=0,1,2, ...
y = 8 lb•s/ft

8. R =10 3 C 2
20. vo < - -y t t 0 /2 m

2. 2n/ 31
23. y = 5 lb . s/ft

4. k =6, r = ±2,is
25. (a)1,715d).73,in r,8 7(e) r = (2/y) In(400/, 4 - y2)26. (a) 11(t) =1 1 0 , 4 k m - 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,=27 3 pll pog
(a) u =f sen f te) horario
(a)=(16/J23)c" sen(1177 t/8)c)
(b) u =os ( N kzt) - v 611 - 7 : sen( 17 f i T I )32. (b) u = se n t, A. T = 27c) A,98, 7' = 6.07(d)E=0,2, A =0.96. T =5,90:  =0,3, A =0,94. T =5.74

(f) e = -0,1, A =1.03. T =6,55; = -0,2, A = 1.06, 7' = 6.90: E = -0.3,
A,11,T =7,41
Seca() 3.8
1. -2 sen 8t sett t. 2 sen(t/2) cos(13t/2)3. 2 cos(37rt/2) cos(nt/2). 2 sen (7t/2) cos(t/2)
u" + 256u = 16 cos 3t.(0) = b,
 
1'(0) = 0, t em ft. t em s
it" + lOu' + 98u = 2 sen (r/2), u(0) = 0,
 '(0) =0,03, u Cm in, t em s
(a) uos 16t +os 3tc) w =16 rad/s
(a) u =153!281[160e75' cos(3 t) +

"- ` se n

) - 160 cos(t/2) +
3128 sen(t/2)]
(b) Os dots primeiros termos sao transientes.

d) co=4./3 rad/s
rt ==" 1
 
(cos 7t '4 2 58 senos 8t) =en (t/2) sen (1502) ft, I erns45It=(cos 8t + sen 8t - 8t cos 8 0/4 ft, t em s;/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,<t <
15. 11 =o[(27r - t) - 3 sett tj, <<7 r
-4F0 se n t,7r < t < oo
' A palavra slug signitica I e s n z a . mas, neste context°, ë 111113 unidade de massa no sistema ing1C's: é uma massa que acelera
1 pc por segundo ao quadrado quando sob a aca o do uma forca de 1 l ibra: 1 slug=14.5939 kg. (N.T.)
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RESPOSTAS DOS PROBLEMAS 567
10-6(e-4"' -3) C, t ems ,(0,001),5468 x 10-6;Q(0.01),9998 x 10 -6 :(t) -* 3 x 10 -6 quando t0(a ) u =[32(2 - (0 2 ) cos w t0)sen wt]/(64 - 63w 2 + 16w4)
(b) A =8/, . /64 - 63(0 2 + 16w 4(d) w=3./14/8,4031, A =64/ ,,/127 I" 5.6791
(a ) a =3(cos t - cos w t ) 1 ( w 2 - 1 )
(a ) tt =[ ( c o 2 + 2) cos t - 3 cos cot]1(w 2 - 1) + sen t
CAPITUL 0 4

eciio 4.1
2. -oo < t <co

t > 0 ou t <0
3. t >1, ou < t <1 . ou t <0

. t > 0
, -37/2 < x < -7/2.7/2 < x < 1. < x < 7/2, r/2 < x <37/2....
-oo < x < -2.2 < x < 2. 2 < x <
Linearmente independente
Linearme nte depende nte: f , ( t) + 3f . ( t)- 2f,(t) = 0
Linearm ente depende nte: 2f,( t) + 13f (t) - 3f ,(t) - 7 f ,(t) = 0
Linearmente independente1. 1
12. 13. -6e-2'
14 . e - 2 (5 .x
16. 6/x7. sen 2  t =
10
(5) - z cos 2t
19. (a) ao[n(n - 1)(n - 2) • • 11+ a l Inut - 1) • • • 2]t +• + a,,t"
(aor" +a1rr.- • •

,,)e'r
e'

e 2 '

im .

` e' e 2 '

, -cc <t <co
21. W(t) = ce - 2 '

2. W (t) =
23. W (t) = c / t 24. W(t) =•/t27. y = ( )e l + c 2 t + este' 28 . y = c I t- + c,t 3 + c3 (1 +1)Seciio 4.21 . f si(:r14)+2.1
3. 3 e i ( T V2m.7).rli3T1214.2nrri
5. 2eilIIIN/61+2rill
., . / e d ( s x / 4 , 1 4 2 / 4 4 : T /
7. 1.(-1 +, ( -1 -. 2r 4 eM i / S , 214eri/8
9. 1 ,

1. -i

./73 +(./. ; + i) I N i:5:11 . y =c 2 re' +=c 1 e' + c 2 ie' + c3t2e'13 . y = c l e r + c,e 2 ' +4. y = c 1 + (7,1 + c 3 e 2 z + c,te2'15 . y =  cos t +en t +e " = (c 3 co s+en e - 1 5 1 1 2 (cs  cos ! t +c t, sen t)y = c, e : +c 3 e 2 ' +c4e2 'y =c 1 e i + c2 te' + c 3 t : e' + c 4 e - ' +c6r2e-'y =c i + c2 t + c 3 e' + c 4 e - : + c 5  cos t + c 8  se n ty =c + c2 e' + c 3 e 2 r - c, cos t + (7 5  se n t20. y =c 1 + c 2 e 2 ' +c 3  cos 0 t + c 4  sen)
y = e'[(c t + c2 t) co s t + (c 3 + c,t) se n II + c`[(cs + ca) cos t + (c 7 + c8 t)sen ti
y = (c 1 + c2 t) co s t + (c; + cot) se n t3. y =c l e f + c2e(2+`f ± c3e(2-.3),
24. y = C2e(-2+,3e(-2- 3 2 ) (
y =

2 + c 2 e - " 3  cos(t/0) + c 3 c -r:3 sen (t/O)
y = c 1 e3' + cze- 2 : +

c4e (3 -A`
y = c I C° + c 2 e - ' 1 43 e - ' cos 2t + c 4 e - ' sen 2ty =c l e - ' cos t  + c 2 e -: se n t + c 3c' cos(0 t) + c4 e - 2 1 se n)29. y =2 -2 cos t + sen t0.=:en(t/./2) - e s / 1 2 se n ( t 1 N / 2 )31. y =2t - 32. y =2 cos t - se n t33. y =e' --- - z / 234=P,e'1 2  co st +1 /2 se n ty- 18e - ' 1 3 + 8e-r2 170- y = 2 ;os t -en t -OS (9 en(f t)
y =cosh t - co s t) +senh r - se n t)
(a) W (1) = c , uma constante (b) W(t) = -8c) W(t) = 4
39. (b) a t =c i co s t + c 2  se n t+c co s .16t+c se nf
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568 RESPOSTAS DOS PROBLEMAS
Sec5° 4.3
I.y ee r + c2 te l + c3 e-' + Ite" + 3
y ce` + c 2 e - ' + c3 co s t + c 4  se n t - 3t -en t
y = C2 COS t + C3 SCI1 t 4(t - 1)
y = ci + c 2 e' + c 3 e" + co s t
y = + c t + c 3 e -2 ( + c4e2i _ 3e- t4
y = c, cos t + c2 se n t + c 3 t cos t c4 t sen 3 + ycos2t
y =+ c t + c 3 1 2 + c 4 e - ' + e' 1 2 [c 5  cos(0 t/2) + c6 sen(0 t/2)] + .; t4
y =+ c t + c 3 t 2  + c 4e +en 2t + 1 6 cos 2t
y =

os 2t) +
10 . y =( - 4) cos t -(Zt+ 4) sen t +3t + 4
I1. y = I + 1(t 2  + 3t) - t e i
49
)=- I co st - en t +e3' Ncos 2t - 1sen 2tY (t) = t(A0t 3 -F A l t 2 + A2t + A 3 ) + B t2e'Y(t)=t(Aot + AOC' + B co s t + C se n t
Y ( t) = A t 2 e r + B co s t +Csen t
16. Y (t) = A t '- + (B ut + B I )e' + t(C cos 2t + D sen 2t)
I 7 . Y ( t ) = t (Aot 2 + A l t + A 2 ) + (B ot + B t ) co s(Cot + C I ) sent
Y (t) = A e` + (B ot +

te' (C co s t +D sen t)
ko = a0,„ = aoa" +

• • • + a„_ a + an
Seciio 4.4
y =+c 2 cost + c 3  se n t - In cos t - (sen t) In(sec t +ta n 1 )
y =c, + c2 e l + c3 e-' - ; 1 2. y =c i e! + c,e" + c 3 e 2 ' +y =+ c2 co st 3 se n t + In(sec t +tan t) - t co s t + (sent) In cos ty =c l e fc co s t + c 3 se n t-, e - ' co sy=c 1 co s t + c 2 se n t + c 3 t co s t + c 4 t se n t - l t 2 se n t
y ce' + c 2  co st +en t -co s t) In cos t + (sen t) In cos t -os t
- 2t sen t + -e l
2
/ Coss I ds
8. y =+

3e-r - In sen t +In(cos t +1) + f (e' 1 sen s) e l s -
1 7 .
+ le" f (e'/ se n s) ds
c l =0, c 2 = 2, c 3 = 1 em resposta ao Problema 4
c 1 =2, c, =3 =4 =em resposta ao Problema 6c 1 =Z. c, =Z, C 3 =to =0 em resposta ao Problema 7c, =3, c2 =0 , c 3 = -e ra , o = :112 em resposta ao Problema 8Y ( x ) = ...r4/15
Y(t)e" - se n (t - s) - cos(t - s)]g(s) ds
• (I)
Y ( t) =senh (t - s) - sen (t - s)]g(s) ds
to
Y(t )

e(-" (t - s)2 g(s)ds; Y(t) =-te t I n 1 1 1
Y(x) = z f [(x / t 2 ) - 2( x 2 /t 3 ) + (x 3 It4 ) ] ,g(t) dr
x o
CAPITULO 5e ca( ) 5.1
1. p= 1
3. p =oo
5. p=i
7. p = 3
f_ltx2n+1
9. E "

(2n + I)!

 = 00n.0
2. p =2
4. p=2
6. p = 1
8. p =e
1 0 . E =!'n =o n
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RESPOSTAS DOS PROBLEMS 569
11. 1+ (x - 1), p = 0 o
o c
1 3. Ec_i>"÷i)"=n=1t i1 5. Ex", p = 1
n=0
12 . 1 - 2 (x + 1 ) + ( x + 1 ) 2, = oo
14. E(-0"x", p = 1
n.0
16. E(-1)"+ 1 (, _ 2)", p =1
n=0
=1+22 x + 3 2 x 2 + 4 2 x 3 ± . . . _2xn
y" =2 2 + 3 2 •2x + 4 2 • 3 x 2 + 5 2 • 4 x 3 + • • + (n + 2) 2 (n 1)x" + • • •
y ' = a l +2a,x + 3a 3 x 2 + 4 a 4 x 3 + • • • + (n + 1 ) a „ . , . ix " + • • •
= Ena„x"- 1 =E(n +1)a,:xn
n= I.0y" =2a, +6a3 x2 a 4 x 2 + 20a 5 x 3 + • • • + (n +2)(n +1)an+,e + • • •o c n(n - 1 ) a „ x " -2 = E(n -2)(n +1)a„_:ann -2=0
ti
x "2 .„ -- 2 x "n=0=2
23 . E(n + 1)an x "4 . 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?

6 . a l +E

1)an. 1 . 1 +a„ - 1  ] • r "
n=0

=1
2 7 .(n +1)fla„ 4 . 1 + a„ I x "8 . a„ = (-2)"aoln!,n = 1 , 2 0e-2'n=0S e c i io 5 . 2I . ( a ) a„ +2 =a„/(n + 2)(n + 1 )x 246x:n(b,d)x) = 1 +• • =E ( 2 1 1 ) 1
n =U
x 3x•
) . 2 (x) =x +
3 !
- +
5 1
- +
7 1
- +E 2 n 4 . 1 ) ! = s e n h5  72 " 4 - I
2 . ( a ) ( 4 . 4 - 2 = 2 )x2 . r 6  2 n(b.d) y1 + - +2. 4 • 6+ ' 2 n n 1n = ox 3 2nn!x?"÷Y z ( x ) =3 - + 3. 72n +1)!3 . ( a ) (n +2 ) a , , f2 - an+1 - a„ = 0( b ) .Y1(0 =1 + . 1 . ( x - 1 ) 2 + 1 ( x - 1 ) 3  + • k ( x - 1 ) 4 + • • •Y 2 ( x ) =1 ) + 1 ( x - 1 ) 2  + 1(x - 1 ) 3  + 1 (x - 1 ) 4 +4 . (a ) -k2 anl(n + 4 ) ( n + 3 ) : a: =a3 =0
k 2 x 4k4.06 x i 2
(b,d) y i (x) = 1
3 -4
+
3 . 4 . 7 . 8. 4 . 7 . 8 - 1 1 • 1 2 +
m=0
Sugestlio: a l g a n = 4n1 n a relacäo de recorrencia.M =1, 2 , 3 , . ..
5 . ( a ) (n + 2)(n + 1 ) a n i - 2 - n(n + 1 ) a n . , . 1 + a„ = 0, n >1; a2 =- la ( )
(b) y i (x) = 1 - -1-x 2 - .1x 3 -• . • , y 2 ( x ) = x - i x 3 --• • •= cosh Xc c :_1)ni--1(k2x4)nt-I1+ 34 7 8 4 34)nt=k 2  X 54 x 96x138 .4• 5 • 8 • 9 • 1 2 • 1 3L [1 + E ( -1) ' " -3 (k2x4) ' • '4 • 5 • 8 • 9 . • • ( 4 n 1 + 4)(4m + 5 )
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570ESPOSTAS DOS PROBLEMAS
(a) a„ . f . 2 = —(n 2 — 2n + 4)an/i2( n + 1)(n +2)1. n2; a2 =— ( 1 0 . a = —
(b) y i (x )— . v 2 +  :46 x 6  • • • .
Y 2 ( x ) = x — 4 Xi+ 1 6 0 x 5 —7 + • • •
(a) an+2-= —an/(n + 1),
. v 2
(b.d) y i (x) = I —
1
n=0.1.2....
x46
. =+ (-1)nx2n
1. 3 1 .)
= 1
1 • 3 • 5 • • • (2n — 1)
X3

5 x. (_1)nx2n+1
Y 2 (X) =X -

2 4 4 - 2 • 4 6 + = n=} 2
4 • 6 • • • (210
(a) (4.4.2 = — [ (n+ 1) 2 an+1 + a„ + a n.-1]1( 1 1)(ri + 2),n =1,2....a2 =—(ao + a l )/ 2(b) y i (x )— 1(x — 1) 2 +(Y - ) 3 -1)4 + • • •Y 2 ( 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 : =0, 1, 2, ...
(b) (x) = 1 — 3x 2 , Y2(x ) =x — x3/3
(a) 4(n + 2)an+2 — (n — 2)a„ = 0: = 0.1.2....
x2357
( b , d ) ))1(x =  —                                                                                                                                                                 y(x) _ 1,240
(a) 3(n +2)an+2 — (n + 1)a„ = 0: = 0.1.2....
x 2

4

6 3 .(2n — I)(b•d) yi(x) = I + 6 +41 432 x 6 + 3 " 2 . 4 • • • ( 2 , 1 )2

1 6 2 (2n)
Y2 (X ) =X 1--X3X5 +935
X 7•945 • x2"13' • 3 . 5 • • (2,1+ I)
(a) (a + 2)(n + 1)an+2 — (n+ )na„_ 1 + (n —1)a„ =0:  =0, 1, 2....
X 2v 3vxt
(11,d) yi(x) =1 +• • • +• • •2(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284"(2n)!
x 3xx4• • (211+ 2) ,
Y2 (X) =
+- -

• • • + (-I)"
 
n (2n + 1)!
(a) 2(n • 2)(n +1)an+2 + 3(tz +1)a„.. 1 + (a +3)a„ =0;  =0, 1, 2....
(b) yi (x) = 1 —

2 ) 2 +

2 ) 3 +

- 2) 4 + • • •
y7(x) = (X. - ) — IOC - 2) 2 +. v — 2 ) 3 + 4-(x - 2) .= + • • •
(a) y =2 + x + X 2 - I -3 +4 + • • -c) cerca de0,7(a) y =—1 + 3x + x 2 — , x 4  • • •c) cerca de lx1 <0.7(a) y— x — 4x 2 + Z x 3 + 3.r 4 + • •c) cerca de lx1 < 0,5(a) y =—3 + 2x — ix2 —• • •c) cerc a ( le Ix ' < 0 .9(a) y i (x) = 1 — .1(x — 1) 3 —x — 1 ) 4 +x — 1 ) 6 + • • •
Y 2(x) = (x — 1 ) — 1(x—1) 4 — .A(.v — 1)5 +(r — 1)7 + • • •
t i(x — 4 )(?. — 4)(A — 8 ) 6
21. (a) y1(x ) = 1 —
2
—
!!4x 2 +1
— 2(X — 2)(X — 6)

2)(X — 6)(X — 10) 7+ x5 + • • •2 (x) = x

!

 !

!
1, x, 1 — 2 x 2 , x — 3x 3 , 1 — 4 x 2 +1 ..v — 1,r 3 +
1, 2x, 4x 2 — 2, 8x 3 — 12x. 16x 4 — 4 8 .v 2 + 12, 32x 5 — 160x 3 + 120x
22. (b) y =x — . v 3 /6 + • ••
Seciio 5.3
I. 0"(0) = — 1 ,:/,'"(0) = 0, 014)(0) = 30"(0) = 0,"'(0) = — 2 , (4 ) (0) = 0
0"(1) = 0,"'(1) = — 6 , (4 ) (1) = 42
0"(0) = 0,"'(0) = — a 0 , 0 1 4 ) (0) = — 4 a 1
p ==00 6. p =1 , p=3. p =
7. p =1 ,= S. p =
4 n (2n — 1)(2n + 1)
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RESPOSTAS DOS FROBLEMAS 571
9 (a= 00b) p = ccc) p = o od ) p =c oe) p = 1(0 p =4g ) p =o ch) p =1i ) P = 11) P = 2( k ) P = 0I ) P =m ) p=o on) p = c oa 22 22 ) a 24 22)(22 a 2 ) a0. (a) y i (x) = 1r z42 !!![ (2ni - 2)2 - a 2 ]22 a 2 ) a 2
( 2 m ) !
1 - a' 33 2 - a 2 ) (1 - a2)Y 2 ( x ) = x 4 - -- +3 !![(2m - 1 ) 2 - 4 2 1 • • • ( 1 - a 2 )( 2 n z + 1 ) !y i (x) ouy 2 (x) termina corn x" dependendo se a = t z è par ou imparn = 0 , y = 1 ;== x: tz = 2. y = I - 2x 2 : n =3, y =x - 1 X 31 1 ' V I ( x ) =f ix5k i x 6 + • • •( x )r 4fix 6Tc i x 7 + • • •p = o oy i ( x ) = 1 - x 3 +(  . 5  + • • • , y 2 (x ) =   t , x 4  +4x6 + • • •p = 00y i (x)= 1 +' +45 x 6 + • • • ,2 (x) = x +kris + 40-x 7x• • • ,
p =7 r / 2
y i ( x ) = 1 + h . Y 3  + 1 x 4 -• • • .2(X) = X - ix 317X5• •
P = 1
N i io é p o s s iv e l e s p e c i fi c a r c o n d i c ö c s i n ic i a is a r b i tr a r ia s e m . v = 0 ; lo g o , .v = 0 c u r n p o n t o s i n g u la r .
x 2n-.y=l+x+- 4••••+-+•=e`2 !!x 246y=1 + - ++• 4• 4 • 6" • n !y = +x +• • • 119. y = I + x + x 2 + • • • + x " + • =,x n3'20 . y = a„ ( I + x + , -• • • +-i + • • .) + 2 ( v+ :,i - , - ,-  • • • +
• 2= me + 2( e' - I - x -
2
= cc ' - 2 - 2x - x=
Y 246 ,- I ) , , x 2 "( 22 2 !3 3 !'n !• )1 . y = ao l - . - -- + 2-- -2-+ • • +. v 2. 34r 5+ ( x + 2- - 3 - 2 -4• 5• • • )= a o c - ' /2 + (x + 2 :) . -33 -2 4 + 3   5- 5. ' ).1. I - 3x 2 , 1 - \\lO x 2 + 3x 4 ; X, x - 3x 3 , X - 13x3 + 4x'(a ) 1 , x , (3x 2 - 1 ) /2 . (5x 3 - 3x)/2 . (35x 4 - 30x 2 + 31/8, (63x 5 - 70 x 3 + I5x ) /8(c) P . 0;2 , ±0,57735; P 3 , 0. ±0,77460;4. ±0,33998. ±0,86114 ;P 5 , 0, ±0,53847, +0,90618
Secijo 5.4
1 . y = c i x2 x -22=c lx +11 - 1 1 2 + C2 IX + 11 - 3 '23 . y = c 1 x 2 + c x 2  I n I x '. y =c i x - ' cos(2 I n lx1) + c 2 x - ' s e n ( 2 I n lx 15. y = c l x + c 2 x I n I x 1. y =c i (x - 043 C 2 (X - 1 ) - 4y =. 2 1 4 - 5 - i f t z b ay = c 1 1 X 1 3 1 2   C O S ( 1 0 In Ix') + c 2 1 x 1  sen (1./3 ' I n l x 1 )y = clx 3 + C 2 X 3 In I• Iy = c i (x - 2 ) -2 c o s ( 2 I n I x - 2 1 ) + c 2 (x - 2) -2 sen(21nIx - 21 )y = I I X I - 1 / 2 cos(2 f 1 . 3 I n l x 1 ) + c 2 1 x 1 " " 2 s e n ( 1, - 43 I n 1 x 1 )y = c l x + c 2 x 413=2 x 3 2 - x-'14 . y = 2 x - "cos(2 In - x - 1 1 2 s e n ( 2 I n x )5. yx 2 - 7 x 2 I n I x !1 6 . y = x - ' c o s ( 2 I n7. x = 0. regula r
18 . x = 0, regular;= 1, irregular9. x = 0 . i r regula r ; x = 1, regularI - x x"n !
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I\ 2 ±11+3) /2 )2!(1+:1i)(2-r
572 R E S P O S T A S D O S P R O B L E M A S
20. x = 0, irregular; x1, regular1. x = 1, regular; x = –1, irregular22. x = 0, regular3. x = –3, regular24. x = 0. –1, regular; .v = 1, irre g ular5. x. regular; x = –2, irregular
26. x = 0, 3, regular7. x =1. –2, regular
28. .1 = 0, regular9. x = 0. irregular
30. x = 0, regular1. x = 0. regular
32. x = 0, ±mr, regular

3. .1 = 0. ±ru , regular
34. x = 0, irregular;

rur, re g ular

5. a <
36. f i >0

7. y = 2
a >
(a) a <1 e0
a < 1eJ3> 0, ou a=efl>0
a> 1 efl > 0
a > /3 > 0, ou1 e > 0
a=0
x -
41. y =ao (1 –
2 • 5 + 2   4•5 • 9
44. Porto singular irregular
46. Ponto singular reg ular
48. Ponto singular irregular
Se(*) 5.5
45. Porto singular regular
47. Porto singular irregular
49. Ponto singular i rregular
an-2
I. (1) r(2r – 1) = 0: a„=1 ==0(n + r)12(a + r) –11v .=v 46x ' 2 [1 — •• 4 • 5 • 9  . 4 . 6 5 • 9 • 13."12"n!S • 9 • 13 . • • (4n + 1) +
.v 4
y2(x) = 1 –
2 • 3
+
2 • 4 • 3 -74. 3 . 7 . 11 ±
(-1),:x2"
+
2"n!3 • 7 ll• • (4n – 1)
a n _ 2
(n + r) 2 –
2. (b) r– =0;an = ri =5 , r =
(c) y 1(x ) =x 1 1 3 [ 1
(-1)"
+
m ! ( 1 - 4 -3)(2+ 0 • • • ( in+1) (2

r 1x \ 2 +X4
y2 (x) =x-I/3 [1
1!(1 — 1) 1 / 2 )2!(1 – 0(2 – 1 ) 2
– 5 )(2 –• • • (In — 1) 1 / 2 . )
t x \ 2, n–iv,
7)7!(1i
Sugesulo: faca a =2m na re laci lo de recorracia , m =1.2, 3, ...
3. (b) r(r – 1) = 0; a

n_ i
„ =

r=1, r =0(a + r)(rz + r – 1 )x`

( c ) Y1(x) =
'x . [ 1 – —!2! - 4 - 2—3-1)"! + * + n!(n + 1)!a„_14.(b) r 2 = 0;„ =i = r, = 0(a ± 62'x2y i (x) = 1 ± —± +_
aw2 ! ) 2 n(n!)2C "
5 (b(3r – 1) = 0; a„=–
n + r)[3(tz + r) – 11:
+ •
=2 =0
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RESPOSTAS DOS PROBLEMAS
(c) Y i(x) =x i" 1 ( x 2 ) 2 + • •1 ! 7! 7 1 3)(-1)'" x2ni!7 • 1 3 • • • (6 m + ) ( 2)•"]
(d ) y 2 (x ) =1 -
2
1
2
) +
1 1 5! 5 •1 22x )2 ( — 1 ) " '
q"'
z !5 • 1 1
(6 m -
1 )
Sagestiio: f a c a n =

a relacao de recorrencia,m =1 . 2 , 3 , . ..
6 . (b) r - 2 = 0;„-12 =(n + r) 2 - 2 .v 2y 1 ( x ) = x 1 2 [1++1 ( 1 + 2 V 2 )! ( 1 + 2 v i ) (2 + 2v2)(-1"x" + • • -]n!(1 + 24)(2 + 2 i2) • • • (n +2 . 4 )x 2y2(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„ =i =r =0x 23(c) y i ( x ) = 1 + x +- - ,• +—,=`8 . (h) 2r 2 + r - 1 =2n +2r - 1)(n r +1 kl„ +( -1)" 'x ' " 'ne7 • 1 1 • • • ( 4m + 3 )( -1)mx2'(d) y 2 (x) =x - ' (1 - X2 + X42 !5 - . . .+m !5 . 9 .. . (4m - 3) ±9. (b) r 2 - 4r + 3 = 0; (n + r - 3)(n + r - 1)a , - (n + r - 2)a„_ 1 =0: r, = 3, r 2 =
2. v
x"
( c ) y l   ( x ) = x ' ( 1 + -3
x + — +
 n!(n + 2) + )
(b) r - r +

; (n + r - ) 2 a . ± a n = r, =1 / 2r 2(c) y1 (x) = x 1 /2 (1-— ++2 42 " ' ( m ! ) 2(a ) r =0; r =0 , r =0CY(a + 1 ) a(a + 1)11 • 2 - a(cr + I)](b) y i ( x ) = 1 + 2 12 ( x)2 1 2 )(2 22) (x) 2 +• • ••_ 1 4 _ 1 y a ( a + 1 ) 1 1 . 2 - a(a + 1)1— [n(n - 1 ) - a(a + 1 ) 1 (x ) "2"(n!):12 . ( a ) r, ==0 em am bos x = ±1
(b)Y I ( x ) = I x - 111/2
x[1+E (-1)"(1 + 2 a ) . • • (2n - 1 + 2 a ) ( 1 - 2a ) . • • (2n - 1 - 2 a ) ( x) " 1
2"(2n + 1)!
Y 2 ( x ) = 1
(-1)"a(1 + a ) • • • (n - 1 + a)(-a)(1 - a) • • • (n - 1 - a )
+x - 1)"n!1 • 3 • 5 • • • ( 2 n - 1 )1 3 . ( b) r =0; 1, r2 = 0; a„ =(n - - ).)an-:n2( -A ) (1 - A ) 2— A ) ( 1 — A.)- • • (n - 1 - A .)(c) y1( x ) =1 + (p2 ! ) 22 x  +++(n)2
P a r a A = n, o s c o e f ic i e n t e s d e t o d o s o s t e r m o s d e p o i s d e x " s a o n u l o s .
16. (c) [(n - 1 ) 2 - 1]!)„= -b„_ 2  e e impossivel determinar b, .
573
2 a , , _ 2 = 0 :
r 1 = =-I Y 2(c) y i ( x ) = x 1 1 2 ( 1 71 7 • 1 1 + • • .)
ti= I
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c c
1. yi(x)=
E_1)"x"
n ! (n+1) !
n= 0
Y 2( X ) =x) In x + [1 - E!(n - 1)! ( 1)".0ii„ + II„ 574 E S P O S T A S D O S P R O B L E M A S S e c ä o 5 . 6(a) x = 0;(a) x = 0;3. (a) x = 0;(a) .v = I; (b) r(r - I) = 0; r 1 = 1 , r , = 0r 2 - 3 r +2 = 0 ; r i2, 2 = I(h) r(r - 1) =0; r 1 = 1 . r 2 = 0(b) r(r + 5) = 0: r 1 = 0 , r, = -5N a b t e rn p o n t o s i n g u la r r e g u la r(a) x = 0;(b) T 2 +2r - 2 = 0 ; r 1 =1 + . 7 = 3. 7 3 2 . r , = -I -= .- -2,73(a ).v = 0:b)r ( r -; r : =2 = 0(a) x = -2;b) r(r - 0 ;1 =2  =(a) x = 0:b) r 2 + 1:t = r , r, = -1S. (a ) x = -1;(h) T 2 -7r t - 3 - -= 0 ; r 1 =7 + . /3 7 ) /2 - 1 ' 6 , 5 4 . r 2  =( 7 - 41 ) /2 1-1' 0 . 4 5 99 (a) x = 1;b) r 2 + r = 0 ;1 = 0, r 2  =-I(a) x = -2:b) r 2 - (5/4)r = 0 :l =5/4 , r 2  =( a ) x = 2 :b) r 2 -2r = 0 ;2, r 2 = 0(a) x = -2;b) r 2 - 2r = 0;1 = 2, r , = 0(a) x = 0;b) r 2 - (5 /3 ) r = 0; r 1 =5/3. r 2  =0(a ) x =-3 ;b) r 2 - (r/.3) - 1 = 0:1 = ( 1 + . 1 3 7 ) /6 " L - 1 , 1 8 ,r , = (I - . ,71)/60 , 8 4 7( b )0 ,2 = 0(c) y 1 ( x ) =1+ x + yx 2 + i; • 3 + • • •y, (x) =3 7 1 ( x ) In x — 2x -• • .(b) r 1 = 1 , r 2 = 0(c) y (x) = x - 4x 2  +  X 4• •Y 2 ( X )6y, (x) In x + I - 33x 2  +"T x 3  + • • •(h) r 1 = 1 , r, =0(c ) Y1 (X ) = -t-1 X 3• • •y 2 ( x ) = 3 y , ( x ) In . v + 1 - 2 .4x 2 - V x 3  + •(b) r 1 = 1 , r 2 = 0( c) y i (x) = x2,1 2 '4 4 "Y 2( X ) = - Y i ( x ) I n . v + I -( b )1 , r 2  =-I( C)Yi(x)=x-13+Y 2 ( x ) =x ) I n x +4 .v 3 + • • •18 . (b) r 1 =2 = 0(c ) y i (x) = (x -1) 1 /2 [1 - i(x -1)+x — 1 ) 2 + - .1. (d) =19. (c) Sug esti lo: (n -1)(n - 2) + (1 +a + f3)(n - 1)+0 = (n - 1 + a)(n - I + / 1 )(d ) Sug estl io: (n - y)(n -1- y)+ (1 + 01(1- y)±0(13 =(n - y +a)(11- y +13)Seca() 5 .7 1 °`' (-1)nx"yi (x) = - Ex n=o(n!)2(-1)"2"yi(n!)2n=0 2N-," ‘ " ( - 1 ("11„ny 2 ( . v ) =x ) I n x ( n ! ) 2n=1E _i)"2"H„Y 2( x ) =x ) I n x - 2 21,1n ! )14 . yi (x)= - Enx! (n+1)!_oY 2 ( x ) =x) I n   x + 11 „+11 „_1".e2 1)!
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7. F(s) = s 2: . > IblS - a
(s - a) 2 - G2'
s 2 + 1)1' s >
0
h13 . 12
(s) =(s - a)' +  b2'
15 . F(s) =
(s -
1
a)-
,

> a
s' + a-
s > 0
9. F(s) =
1 1 .s)
17. F(s) =
- a) 2 (s+ (1)2'
2a(3s : -
19. F(s) =s- +s >
s - a > I b l
s > a
RESPO STAS DOS PR OBLEMAS 575
5. YI(x ) == A 3I2[1_ 1 . ) m„,. 1 101 + N2+ 0 • • • (In +
[
(:_ • \ 21 ,1]
Y2(X ) =V-312 1 +E
!(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

eciio 6.1
I . Seccionalmente continua
3. Continua
(a) F(s) = 1 / . 5 2 ,
(c) F(s) =tz!/sn+1,
F(s) = s gs22) ,
8. F (s) =
s 2 -b- b 2>
10. F(s) =
s - a) 2 - h2
b
a > I b l
12. F(s) = s
> 0s2214 . F(s) =- a> a(s -a)2+ b'16 . F(s) =as,„> 0(s- + a- )-n!
20. F (s) =2a(3s2
2)18 . I; ( S) =
(s2 - a2)3
on+, >a
> lal
2. Nenhuma das duns
4. Seccionalmente continua
(b) F(s) =2 s 3s >0
s > 0
s > 0
21. Converge2. Converge
23. Diverge4. Converge
26. (d) 1(3/2) =2: 11/2) =2
Secao 6.2
1. f (t) =; sen 2t
3. f(t)= te-415. f (t) =2e" co s 2t7. f (t) =2e.' co s t + 3e' se n
9. f (t) =-2e -2( co s t + 5e-:' se n t
11 . y(e3r + 4e-2`)
13. y = e l  set) t
15 . y=2e' co s- (2/ ifl)e t se n ./5
17 . y =te` - t 2 e : + 313e'
19. .v = cos N i f t
21.y=co s t - 2 sen

e: cos t -2e`sent)
23. y =2e -' te" + 212e-1
1

- ̀(s + 1 )
25. Y ( s) =
2. f (t) =2t2e'
4. f (t) = e 3 ' +
6. f (t) =2 cosh 2t - sehn 2t
8. f(t) =3 - 2 sen 2t + 5 cos 2t
10 . f (t) =2e' cos 3t -en 3t12 . y =e-2:14. y =e:r - te2'
16 . y =2e - t cos 2t + ; e - ` sen 2t
18 . y =cosh t
20 . y=)w 2 - 4) - '[(w 2 - 5) cos cot + cos 2t]
22. y=i(e - ` - e 2  cos t + 7e sen t)
S

-
24. Y(s) -

+

2 + 4

(s 2 + 4)
26. Y(s) =(1 - e-
30. F(s) =2b(3s 2 - b2)/(s2 + b2)3
32. F(s) = n!/ (s - art
F(s) =[(s - a) 2 - h '1/[( s - a)- + b2]2
a (a + 1)1Y = -1
s 2 ( s2 + 1 )2 (s : + 1)29. F(s) =1/(s - a)231. F(s) = rz!/ sn +1
33. F(s) =2b(s - a)/[(s - a) : + b 2 J 234
36. (a) Y' + s 2 Y = sb) s 2 Y" + 2sY' - [s 2 +
c)/s2(s2 + 4)
Seciio 6.3
(b) f (t) =-2u 3 (t) + 4u5 (t) - u7(t)
(b) f (t) =
1 - 2uz (1) + 2u 2 (t) - 2u3 (t) + u4(t)
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36. EV - e-s=> 0
5(1 + e-5)
1 +
C"
38. r(f (0) =
1 + s 2 )( 1 - e- T '
s > 0
5 7 6 RES P OS TAS D O S P R O B L E M S
9. (h) f(t ) =1 + 1 1 2 (0(e -
( ' -2) - ]
(b) f (t) =t - u 1 (t ) - u (t) - 1 1 3 (t)(t - 2)
(b) f (t) + 1(2(0(2 - t) + 11 5 (0(5 - t) -
13. F(s) =2e -s 1 s '
e -" -2"
15. Fs) - -1 + 7s )
s2
17. F(s) = S
- s 2
2 ((1-s)e - 2 4 - 1 + s)C3s]
19 . f (t) = t 3  e2'
21 . f (t) =2u 2 (t)e' cos(t - 2)
23. f (t) =it i (t)e2 ( 1- 1 ) cosh(t - 1)
26 .f =(2t)"
28 . f(t) =e'1 3 (01 3 -1 )
30. F(s) = s- 1 (1 - Cs), > 0
1
32. F(s) = - [1 - e S + • • • + e -2 "s - e-(2n+11
10. (b) f (t) =r 2 + u (t)(1 - 12)
u 7 (t)(7 -
14. F(s) =e - s (s 2 + 2)/s3
16. F(s) =1 (e' + 2e - 3 s - 6e-4s)
18. F(s) =(1 - e-s)/s2
20. f (t) = 1020[e - 2e-2(1-211
22. f (t)

1 2 (r) senh 2(t - 2)
24. f (t)

t i (t) + " 2 ( t) - 11 3 (0 - 114(t)
27. f(t) =
t •
cos t
29. f(t) =1 2 ( t 1 2 )31. F(s) =s -1 (1 - e -' + e -2s -> 01-(2n-,21.5s(1 + e- s )  s > 0
33. F(s).(-1)"/ s> 0
1 + cs
n=0
1/s
35.4f s > 0
1 + e-s
1 - (1 +s)e-`37. Cif

s > 0sz(1 _ e-s)
39. (a) ,C{ (0) =s- 1 (1 - e -5 ) .> 0r(g(t)) = S-2 (1 - e'),> U.C{h(t)} = S-2 (1 - e -') 2,> 0- e- 5
40. (b) .4)(0) => 0s 2 (1 + e-5)Sectio 6.4
(a) y =1 - cos ten t - u 3 T (t)(1  cos t)
(a) y =e-' se n t + lu,(t)(1 + e - " -') co s t +ell ti
-tt„(t)rt - e - ( ' -2 " ) co s 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

 le2 1 - e-' - t€ 1 0 (011,

e u-IN
(a) y =e -' -e -2 1 + u 2 (t)(1, - e -u -2 ' + ;e-2"
(a) y =co s t +t)(1 - cos(t - 37)1(a) y =h(t) - tt, 1 2 (t)h(t - r/2). 11 (t) =-4 + St + 4e - ' t 2 cos t - 3co sen 0(a) yen t +ite,(0[t - 6 - sen(t - 6)](a) y =h(t) + it, (t)h(t - 7),i(t)=(-4co s t + se n t +4e -o co s t +c se n t I(a) y =11,(t)[; -os(21 -] - 1 1 3 , os(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),(t) =(3 - 4 cos t + cos 20/12
f (r) =(110)(t - to) - 11 z o + k (1)(t -to - k)1(11/ k)
g(t) = (11 1 0 (t)(t - to) - 2 1 1 1 0 +k  (0(1 - to - k) + u,, !-2k ( 1 )  (t - to - 2k)](h/k)
(b) u(t) =4ktt3 1 1 2(t)h(t -4 k 1 1 51 2 ( t )11 (1 - 1 ) ,
h(t) =( 7/84) e- lis sen(30 t/8) -cos(3 ‘17 t/8)(d) k = 2,51e ) r =25,6773(a) k(b) yus(t)h(t - 5) - u 5 +1 ,(t)h(t - 5 - k ))1 k,(t) =asen 2t(b) fk (t )t r 4 _ k (t) - 4+k(t)112k;y =Itt.t_k(t)h(t - 4 + k) - 114 k(t ) I 1 (t - 4 - k))12kh(t) =a -IC ` i6  cos( 143 t/6) - (.543/572) e -0 6 sen(,/iT3 t/6)19. (b) y =1 - cos t + 2 E ( - o k u k , (t)i 1 - cos(( - k7))k=l
21. (b) y- cos t + E(-1) k uk, (I) (I - cos(t -1
k=1
!I
23. (a) y

- cos t +2 E (-1) k uilk/4(o[ - cos(t - tik/4)]k = 1
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RESPOSTAS DOS PROBLEMAS 577
Secau 6.5
(a) y = c' co s t + c  se n t + u, (t)e -(' se n (1 - jr )
(a) y= Itt,(t)sen2(t - 7) - Itt,„(t) sen2(t - 27)
(a) y =115(t)[-e-20-5,-u 1 1 1 0 ( 0 [1 +- 2 ( t - - 1 0 )
(a) y = cosh(t) 2044 3 (0 senh (t - 3)
(a) y =en t - i co s t + le' cos Nr2 t + (11 ./ .) u 3 , ( t) e - ( 1 - 3 T ) sen 12 (t - 37)
(a)
y = z
cos 2t + tt
4 n (t) scn 2( t - 47)(a) y -= sc n t + u 2 ,(t) se n (t - 27)
(a) y =u, f 4 (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) 11,1 6 (t) exp[ -t - 7/6)]sen(OT/4)(t - 7/6)
11. (a) y = 1  cos t +1en t - 5-e"tosost -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- 2 7 4 8(t - 5 - T),= 87/(a) y = (4/./71 7 5 ) t t 1 (t)e -" - 1 1 t 4 scn(./75/4)(t - 1)t 1 L 2,3613, y ) .1  0,71153
y = (8 . /7/21) u i (t)e -"-10sen (3018) (t - 1 ) ; i L 2,4569, y L L 0,83351
(d )1 -F 7r/2 -24 2,5708,i = 1
(a) k l IL 2,8108b) k 1 L 2,3995c) k 1 = 2
(a) 0(t, k) = E t t ,s _ k (t)h(t - 4 + k) - 1 1 4 . 1 4 ( 0 1 ( t - 4 - k))/2k, h(t) = 1 - cost
(b) 0„(t) = u 4 ( t) sen(t - 4)

c) S im
20

0
17. (b) y =E u k , ( t) sen( t - k7 r )

S. (b) y = E(-1)"I li k ,(i)sen(1 - k7r)
k=1=120019. (b) y =E u k , a(t)sen (t - k7r /2)0. (b) y = E(-1)"t i k ,1 2 (t)sen(t - k7/2)14=1=11 5021. ( b) y =E u ( 2 k _ i) ,(t)sen[t - (2k - 1)7]2. (b) y = E(-0kf.,,,,,i4(t)scn(t - 11k/4)k = 1=17 ..0
(b) y=*9-1)"1/4,(t)e-(1-icro0senk/399(t - k7)/20)
k=i
15
(b)V=-E01 ( 2 k _ t)e - [ 4 - ( 2 k - t ) . , 4 1 / 2 0 sen{s - (2k - 1)7)/20)
. 1 3 9 9
Se c ão 6.6
se n t * se n t =; ( sc n t - t co s t) e negativo quando t =27, por exemplo.
F(s) = 2/s 2 (s 2 + 4). F(s) = 1 / ( s + 1)(s 2 + 1)6. F(s) = 1 /s 2 (s - 1 ). F(s) = s/(s2 + 1 ) 29. f(t) = f e - ( 1 - T ) cos 2r d r. f (t) =t - r) sen r d r10. f(t) =  (t - r)e -( ` - ` ) sen 2r dr1. f(t) = f sen([ - r)g(r) dr
(In+ 1 ) 1 ( n + 1 )
P. (c)
I
u m (1 - u ) " du =
1(m + n + 2)
i
13. y= -1 sen (ot + - Ienw(t - r)g(r) dr 14. v=

' -`) sen(t - r) sen a r dr
to

o 0

y = y-(`- ` ) /2 sen 2(t - r)g(r) dr0 rv = e" 1 2  co s t - le -0 2 sen t +( ` - ' ) /2 sen(t - r)[l. - u, (0] drof 4v = 2e -2 ' + 1 e-2r +t - r)e-2('-')g(r) d r
v = 2e - ' - e - 2 ' + e - 2 ( ' - ' ) ]  cos ar dr
1 f '
2
9. y = -senh(t - r) - sen(t - rflg(r) dr
'
k=1
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578 E S P O S T A S D O S F R O B L E MA S
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) = 1,

'(3) = 0
(a) 0(t) = cosh(t)
(b) 0"( t ) - 0(t) = 0,

(0) = 1,

'(0) = 0
(a) 0(t) = (1 - 2t + t 2 ) e - g
(b) 0"( t ) + 2 0 ' ( t) + 0 ( t ) =

(0) = 1,"(0) = -3(a) 0 ( t ) = e 2 cos(,75t/2) + 1- . e/ 2 sen(Ot/2)(b) 0-(t)+0(t)=0,(0) = 0,'(0) = 0,"(0) = 1(a) 0(0 = cos t(b) 0 ( 4 )(t)- 0(t) =0,(0) = 1,'(0) = 0,"(0) = -1,"'(0) = 028. (a) 0(t) = 1 - .e-1/2sen(0/12)
(b) 0'(t) + 0"(t) + 0'(t) = 0,o ) = 1 ,' (o) = -1 ,"(0) = 1
Seca() 7.1
2.

x_ ,

2 = -2x, - 0,5x 2 + 3 sen t

1. x', = x2 ,' = -2x, - 0,5x2
3. x'12 ,' = -(1 - 0,25t -2 )x 1 - t-l x 2 4. x1 = x2 , ' = x 3 , x3 = x 4 , x4 = x,x = x 2 ,' = -4x 1 - 0,25x 2 + 2 cos 3t, , (0) = 1. 2 (0) = -2
= x 2 ,' = -( t ) x 1 - p ( t ) x 2  + g(t);1 (0) = u 0 . 2 (0) = t t ' o
7 (ax= c 2 e - 3 ` ,2 = c l e' - c2e-3 'c, = 5/2,2 = -1/2 na solucâo em (a)
0 grafico se aproxima da origem no primeiro quadrante tangente a reta x, x,.
8. (a) x'' - xi - 2x 1 = 0
(h) x= 1 1 - e 2 ' - ie- , 2 = e2 ' - le'
(c) 0 grafico c assintOtico a reta x, = 2x, no primeiro quadrante .

9. (a) 24
x', + 2x, = 0
x, = - ie/2 _ l e 2 f ,x2 =

2r
0 grafico a assintOtico a reta x, = x 2 no terceiro quadrante.
10. (a) xi + 3x1 + 2x, = 0
x 1 = -7e - ` + 6 e 2 r ,2 = -7e'` + 9e'
0 graf ico se aproxima da origem no terceiro quadrante tangente a reta x, = x 2 .
11. (a) x ' , ' + 4x, = 0
x 1 = 3 cos 2t + 4 sen 2 t, x 2 = -3 sen 2t + 4 cos 2t
0 grafico 6 urn circulo centrado na origem corn raio 5 percorrido no sentido horario.
12. (a) x ' , ' + x ' , + 4.25x, = 0
x l = -2e - ' 1 2 cos 2t + 2e -o sen 2t,
 
2 = 2 e - ' 1 2   cos 2t + 2e -ra sen 2t
0 grafico a urna espiral se aproxima ndo da origem no sentido horario.
13. LRCI" + LI' + RI = 0
18 _V= y3,

 Y4, tn i y; = -( k 1 + k 2 )yi + k2y2
rn 2 y 4 = k2y1 - (k2 + k3)y 2 + F2( t)
22. (a) Qi = i - -,1 , 5 Q, + Q2, Qi (0) = 25
( 2 '2 = 3 + ;WI - 5Q2,

2(0) = 1 5
Qi = 42,i =36
= qx, + x ,1 (0) = -17
x2 =1X 2, 2( 0 ) =-21
23. (a) Qi = 3 t/i - is Qt + yro Q2, Qi (0) =
Q' = ( 1 2 + 3 a Qt — 1 0 o Q2, Q2(0 ) =
Qi = 6(9q1 + (72),f = 20(3q 1 + 2q2)
Não
( d) 192 < QPQ <
+ Fl(t),
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RESPOSTAS DOS PROBLEMAS 579
Seciio 7.266
1. (a)5
2
( c )
612
4
9

2
3
— 2
8
3
7
0
(b)
(d)
—1512
718
—263
— 891
1425
5

8

— 1
— 5
2.a) ( —  i
—7 + 2i) th \ ( 3 + 4i 6i
—1 +2i 2 + 3i) '

1 + 6i 6 — 5 0
(c )
(-3 + 5i
2+i
7 + 5i)
7+2i
(d) (8 + 7 i
6 — 4i
4 — 4i
— 4 )
— 2 1 1— 23.a) (0 2)—1 (b) 21 12 — 3 1 31 0
— 1)
(c). (d)3l
54
3-2i— i) (b)— i )3 + 2ic) (3+ 2i4. (a)
1 + i

2 + 3i

+ i

2 — 3i)

c ' 1 — i
(10
1 40 )
5

4
71 13)(016. (a)107h)— 4121)
2 + i
—2 — 3i)
9c )

6
— 5
8.a) 4i
( 2_1110.
— 8
1 5
— 1
(b) 12 — 8 i
. . ._1.)
1
— 1 1
6
5
(c ) 2 + 2id)6
1
1 1 .
li
1 1 —2
1 — 3 2 3 13
P.3( 3 — 1 113.(3 — 1 0)32 — 1 0 1
3 0 3
1 1 1
2
—s
8
14. Singular 15.( 1 _
4
)
0 0
1
2
1 3
1 0 1 0 1 0
16.. (-1
1 0
4
1 0
1)
to
17.ingular
_ 1 31 0 1 0 1 0
/1 0 1
6 y _ 5 i
5
18 .
0
1
1
1
1
1
)
)
19.
5
0
1 1
--
— 1
—
6
3
1
4
3
1
0 1 0 1
— 2 — 5
5
4
5
5
_ 1
5
5e' 10e2' 2 e 2 1 — 2 + 3e3' 1 + 4e -2 '
21 .a)
(7e
— e t 7e' 2e 2 1(b
)
4e" — 1 — 3e" 2 + 2e-'
—'e 3 ' + 2e' —4')+ te' + e t + e'r8e' 0 — e 2 r —2e ' — 3 + 6e' — 1 + 6e.- 2 : — 2e!3 e 3 ` + 3e' — 2e4'
e t —2e-' 2e2' 1 2e-1
( c ) 2e ' —2e 2 ' (d) ( e — 1) 2 — + +e 1 1 ) )
— e —3e 4e2' — 1 3e -I e +1
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580RESFOSTAS DOS PROBLEMAS
Secao 7.3
1. x 1 = 7X2 =1 , 3 —
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 solucao
c, onde c e arbitrario
onde c é arbitrario
6. x 1 = c 1, 2 = c2, X3 =c2 + 2
8. x ' ' ' —5x( 2 ) +2x( 3 ) =0
10. Linearmente independente
13. 3x")(t) — 6,C 2) (t) + x (3 ) (t) = 0
14. Linearm ente independente

6. A l = 2, x (1 ' = (13) ; A2 = 4 , x (2 )  (11)
1
A2 = 1 — 2i, x (2) =l = 1 + 2i, x ( 1 ) = ( 1 —1A l= —3, x ( 1 ) —)., 2 =  1 , x(2 '  C i )
A 1 = 0, x (1) =2 =2, xa)
)
A l = 2, x" ) = ( A 2 = —2 . x121=
(

•l = —1/2, x( 1 ) =)(I()'2 =
—3/2 x (2) = ()
2 0 0
; ‘ , 1 = 1. x( 1 ) =
(
A2 = 1+ 2i, x( 2 ) = 1) :  1 — 2i, x( 3 ) = (12 i1 0
23. A l = 1, x (1) =
( — 1
; A 2 = 2, x(2 ) = 1) ; 3 =3. x 1 3 )
0
= (

1
2 2
14. A 1 = 1, x( 1 ) =2) :( -1 A2 =2, x( 2) = (1) :, x ( 3 ) 2(_)1 25. —1, x (1) = -4 1 A 21, X' 2 ' =— 1 : + . 1  8 , 7e 3) =(1)2
Secäo 7.4
2. (c) W(t) = c exp iPti(t ) +P22(01 dr
6. (a) W(t) = 1 2
Y e ( ) e x (2) silo linearme nte independentes em todos os pontos, exce to em t = 0; eles silo linearme nte
independentes ern todos os intervalos.
Pelo menos um coeficiente tern clue ser descontfnuo em t = 0.
(0


xd) x =
— 2r))
7. (a) W(t) = 1(1 — 2)es
x") e x( 2 ) sac) l inearmente independe ntes em todos os pontos, exceto em t = 0 e t = 2; cies silo linear-
mente indepe ndentes cm todos os intervalos.
Pelo menos urn coeficiente tern clue ser descontinuo em t = 0 e em t = 2.
((d) x' = 2 — 2t2 — 2 x
t 2 — 2t2 — 2 t
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1 1
x = c 1-4e -: + c 2 0
1 -1
x
= c•;
4 e _ , , _ r ( . 2 _ .4
-7 _ _ . /
1 1
1 4 . x = c l -4 e t + c 2 - 1
-1 -1
15. x = - 3 (3
/ e
4 (/)   e 2 1 + - (I1)
0 1
1 7 . x= -2 e' +2 12 '
1 0
20. x =
1

1 I t +

3
)t-1
-
1
2 2 . x = c l ( 3 ) + c ,

- - ,42) 2e - - ` + c 3' l'?c_, + c 31 e,,-11c - 2 1 + c3ar1 1 6 . x = 118. x = 6 1 ) e,(12 + ,. 03 ,_ W 1e t + 3 2 e - 1 - 1 eit-1 1 -821. x =3)122 3 . x =c: ( 12 ) (2)2) RESPOSTAS DOS PROBLEMAS 581S e c S o 7 . 5 2 ?. (a ) x = c i (1 ) e -: + c 2 (2 ) e- 2 d. (a ) x = c ) (2 )e" + c 2  ( 1 ) e 2 t 13. (a ) x = ( - I CI ) e i + c 2  ( 3 )e'. (a) x = c i ( -4 )e- 3 1 ± C2( 1 ) 2 t15. (a) x = e l (2) e - 3 '  + C2 C) - t. (a) x = c 1 (_ i)e + c2 ( )  e -,,7. (a ) x = c 1 (4) + c 2 (2) C - 2 (. (a) x =- c 1 (-1 ) ÷c2(-31)e`1 .C 1 (2 i\ e; _ m i\ _ „9. x = el()iic 2 ( ) e V0. X -1 )--A-lr1 111 . x -i (1) e 4 ' + (-2)e`1 1 1+ c 3 ( 0 ) e'-1( a )x ,,e =-(c/a)x 1 - ( b / a ) x z(a ) x = e - • +5 C) ti.202 9)  e14(c) 7'4 . 3 9
3 1 . (a ) x = c 1 e ` - 2 + 4 ) 0 2 C2 (4) e1-2-4 )//2.1
r1.2= (-2 ± 4)/2: no
1
x = c 1 ( -1) e ( - 1 + 1 2 1 1 +e 2 (7 2)

1 . 2 = - 1  ±

onto de s e l a
= -1 ± „AZ, a = 1
32. (a ) (v ) =c l() e- 2 '  + C2( 11 ) e - t
Sectio 7.6
33. (a ) ( 
CR2 L)L
2 t
1. (a ) x = clef (c o s 2 t
c o s
+sen 2 t) + c z e t ( - cos 2/
+ s e n 2 t
sen 2/
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COS —sent
10. x = e2t ( cos — 5 Sent
—2 cos t — 3sen t
. x = e'
( cos t — 3 sen
582RESPOSTAS DOS PROBLEMS
2 cos 2t
(a) x = cie" (
se n 2
+ c2e-t —
2 sen 2t
cos 2t
5 cos tse n t
(a) x = (2 cos t + sent
+ C 2
( 
— c o s t +2 sent)
5 sen it
(a) =cie'12 3(cos 5 cos
it
+ sen it))

e
c1 2 (
3( — C O S i t +sen it)
cos t
(a) x = cle

"

en
+QC (
—  cos t +
t
2sen tcos t +sent
( —2 cos 312 sen 3t(a) x = c ircos 3t + 3 sen3ten 3t — 3 cos 3t)0x = c l (-3) e t + c2 e cos 2t) + c 3 e 1en 2t
sen 2tcos 2t
sen f tcos
8. x c (-2) P -2'os f tc 3 een ./2 t
1

cos 4 r — , r 2 - sen

./1 cos4sen t
11. (a) r =

2. (a) r =(a)r a ib) a =0(a) r =(a ±20)/2b) a =, . /Y)15. (a) r =± 14 — 5ab) a = 4/56. (a) r =1 ± 107eb) a =0, 25/12(a ) r =—1+ V:-T eb) a = —1, 0(a ) r = 2./49 — 24ab) a =2, 49/24(a ) r— 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 InI)) 2t

OS(I In t))
(

cos(ln t)

sen(ln 0
"Y?X = C1F C2 cos(ln t) +sen(In t))(— cos(In t ) + 2 ser(In t))23. (a) r = — 1  ± i, -- .14. (a) r = — 1  ± i,(25. n 1 =_i n ( cos0/2) ) + . _ 02 ( sen(t/2)( )/ )lesen(t/2)'e4 cos(t/2))Use c 1 =2 2 = — i na resposta do i tem (b).
lim 1(t) =lim V ( t ) = 0; nao
I-.:"C.-,•W
26. (b) (v) =cie-'
( cos tent
+ c,e"
—
COSI — 
se n
tsen t + cos t)
U se c1 =2 e c2 =3 na resposta do item (b).
lira 1(t) =lim V (t) = 0; nä°
(b) r = ±i11 71 7n

d) Irl e a frequencia natural.
(c) r i = —1,) =3 ) ; r4,2)=(3)— 4x l = 3c 1 cos t +3c 2 sen t + 3c 3 cos 2t + 3c 4 sen 2t,
x, =2c 1 cos t 2c se n t — 4c 3 cos 2t — 4c., sen2t
x'1 = —3c 1 se n t +3c 2 co s t — 6c 3  sen 2t + 6c 4 cos 2t,
x' = — 2c 1 se n tc 2 co s t + 8c 3  sen 2t — 8c 4 cos 2t to
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(4) = (1)
cos 13 t \
— c os 0 I
—0 sen0 t
.173 sen13 t /
sen 13 t
(—en 0 t
± C4 Nij COS N73 t
— 0 cos 13 t
RESPOSTAS DOS PROBLEMAS 583
0 0 1 0
30. (a) A = ( °4
0
3
0
0
1
0
9/ 4 —13/4 0 0
1\
(b) r 1 =i.1) = ( . ) : r2 = — i ,2) =
—i
i —i )
4
r3 = i i,3 )— it I
4 =
— li ' (4) =

103
— 3
1 0 i1 5 •
4
2
(5( cos t \cos t
(c) y = elsent
sent j
(e) c 1 = 10 C2 = 0, C3 =4 = 0.
0 0
31. (a) A= (
0
— 2
11
0

0 )
1
0
1
1
(b ) = (
i
r2
1
— 1
r 3 =0i. ( 3 ) —
A) ;
(
r4 =
—vi
/co s P\sell; t \—3 cos P3 s e r i i t+c3+4—10sen .. t0 cos Z t1 55\ sentt /— TCOS 1 Jper iod° = 4n.
(2) =
+ C2 (
sent
sent
COS I
cost
— 0i,
(c) y = c 1(
cos t \
cos t
cos t
sent J 4 - c '2
 
costt

c3
(e) c, =1. C 2 = 0, C3 = —2, c 4 =0.
Sec:10 7.7
=
)
1 .b) ( I)(t)+ 3 e 2 'e-re2r32e - Ie - , 3 1 e 2 tli e -s/2 + -;e"
(—t/2 —e2. (b) (NO =
1 e - g / 2 — l e - 'e-0 + Ict4
(b) OM .--- - ifre f — 1 e-'

2-
— C'
2

1e` + le'
1e 2-

- I
(b) ^(t) =

e-31 + ie2r

-
_

1 6 , 2 ,
5
- 3 t  e 2 t5. (b) ^(t) = cos t + 2 seri t5 sen tsenos t — 2 sen( e" cos 2t2e' sencl)(t) =en 2t2' cos 2t)
1e 2 ' + e2 r2 e a ,
(N O =
2-+e —:e-2 e(b)(b)
A L _
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584 RESPOSTAS DOS PROBLEMAS
8. (b) (1)( t)
(e - 1 cos t + 2e' sen t
5e" s e n r
e'sent
e -t c o s t — 2 e - ` s e n t
( — 2 e -2 ( + 3 e - 'e - 2 1 + e - 1e-2t1
eee 5l tI-4  -F- - 43—,e12 e: e -D — .4 e' + le
Z
e
-2 ' —  2 e -
' — 2 e 2 c'(b) 4 ) ( r ) - =2 i- 12(72ttt4 -e-2
1 _

e-21— ie . - 1 — -- . e 2 1
2
e' + l e - 2 ' + 1 e 3 '
6

^(t) =

d e r — le - ' + e 3 '
1rt— 6 e— -3 e - + -2 e1 e ' + 1 e-2t)ie . ' — 11-2(r2te t — e -2t — 1 e 3 1(b) — 2 e ' + e- 2 1  +  e 3 (1 t21 .  1—e+ e —.-e -, e — -3 e1 1 . x = —7 1I ) e ` — 2 ( 3 ) e-t2 . x = ( 3 ) e - ' c o s 2 t +' sen2t17. (c) x ( /to ) c o s wt +oe n w tv o( 0 2 1 1 0
S e c a ( ) 7 .8
(c) x =1) e t + c 2 [(I) te t +(0 ) et](c ) x = c i ( , ) ) + c 2 [0) r — (?)](c ) x= c 1 ( I ) e - r + R I )(2)e-](c) x = c i ( i ) e - 1 / 2 + c 2 [( 1 ) te (1 -, +() _ ,]. . ,— 3
5 . x = c i4- ' + c22' + c .e' +) e2'2(— 10— 11(
1((1x = c i12 ' + c 2)e- 1 + c 3- `11- 1( a ) x = (3 -I- 4t) e2 + 4 t— 2. ( a ) x =e r t 2 + 2 (-1 ) t e ( 1 2( — 1 )0
24e' + 32 '
— 6)
41(a ) x = _ _i) e - ' / 2 +1() e-71/237x = ( I ) t + c 2 [( 1 In t + (0 ) t]11 4 . x =I ) c-,[()1 t-- In tI ) t- 1
16 . (b) ( 1 , ) = —e-r '2 + [ (_ 2 ) to - 1 1 2 + (0 ) Cr'8 . ( a ) x =) e — 6 ( 1 1 )10. (a)21 4)  r1 1 . ( a ) x =
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R ESPOSTAS DOS P ROBLEMAS 585
( 0
17. (b) x o )(t) =
 
2'
— 1
(1
(c) x( 2 )(1)= e 2 ( +2(
— I
( 0

2d) x( 3 ) (t) =

 1 2 /2)0 +  e 2 ' +  2 (
— 1

)
0
(1
+2
e 2(e) li(t)= '+ 112/2)+r— 1t(12/2)+30 1 ') — 3 3 2(f) T= (1 0, T = 3 — 2 — 2
— 1 3 — 1 1 1
1
J=0
0
(1

18. (a) x 1 1 1 (t) = ',

(21(t)=

t
— 3
(d) x (3 ) (t)= 4
_.1
0
t e t +— 1 er1: 1 2 2t(e) l“n = e :t( ou e' 0 4 4t
232: — 1 2 — 2 —2t — 1
2—1/2 0
(f) T=
(1
4,-1= 1/ 4 0.
2 _2—3/2 — 1
1

J = (0

0
2x19. (a) J(2k2 )1 ' J3 = ;. 33i.2;.4.1004;0)A4exp(Jt) = e'•' 0 1 )
x = exp(Jt)x°
1 0 0 1 t2/2
20. (c) exp(Jt) = e:" (0
1
t 21. (c) exp(J0 =e' ' ( 1 t
0 0 1 0 0 1
Secao 7.9
x = c i ( 1 ) t + c2 (3) e +; () te g — 4 (3)e' +(2) t — (?)
x = c 1 ( 3e'`
+c ( e - 2 r — ( /3 ) ( —1
-n /-1 Nij C , +/0)e_i
x = ci
s t +sen t4 (— 5sentCOS I + 2sen tc2cost — (1) tsent —1 cost1
x = c l (2) -3 ' + C2 (1) e 2I — C O e-2( +0) et
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C/2 cos
13. (a) 1 1 (1) =4e - Y 2 sen i t2
en I t
-4e-0 2  cos I t
(b) x = C-f/2
4 4 cos It
sen t
586RESPOSTAS DOS PROBLEMS
x = ( 12) + C2 [(2) (
I
)] -2 (2 ) In t + (2 ) t -1--2
x =+ c(- 2
1 ) e _ 5 , + (21) in t 38 (21)t s(-20
2
1 1
7. x = c 1 (
1
) e 3 ( + c2 (
-2
) e +
4( )
1 \ ±2 (
1
1) ter
8. x = c () et + c 2 ( 1 ) e -t + (o)
-4 -1V - 1(+1e,x = c i ( 7,) e' + c 2 (e
v L. 2 - A te- + 9 -1 - -./2
x = C1cost5/2)sen t0+ c '.+ 1/2) t cost -sent - (5/2) c o s t()cost +sent C O S t ± scot
12. x = [-I In(sent) - In(- cost) - 3t + 6.11 (
/ cost +sent)
2cost
+ G- In(sent)- lt + c2] (

sent
-cost 4- 2scnt
t. x=c 1 ( 1 )e-r2 +c2U± (3) - (5
7 ) + \1/
1I;6 t_ 2
CAPiTUL 0 8
1  3i)tInc () t + c  ( 1 ) -I - (2) 21 ) t - I13) t
x c ( ) t 2 + C 2 ( 1 ) t-i-2)  4- (2)
\ 2
 (2)0 \ I)
Seca() 8.1
1 (a) 1,1975; 1.38549 ; 1,56491: 1,73658
1,19631; 1,38335; 1,56200: 1,73308
1,19297; 1,37730; 1,55378; 1,72316
(d) 1.19405; 1,37925; 1,55644; 1,72638
2. (a) 1 ,59980; 1 .29288 ; 1 ,07242; 0 ,930175
1,61124; 1,31361; 1,10012; 0,962552
1,64337; 1,37164; 1,17763; 1,05334
(d) 1,63301; 1,35295; 1,15267; 1,02407
3. (a) 1,2025; 1,41603; 1,64289; 1,88 590
1.20388; 1,41936; 1,64896; 1,89572
1,20864; 1,43104; 1,67042;.93076
(d) 1,20693; 1,42683; 1,66265;,91802
4. (a) 1,10244; 1,21426; 1,33484; 1,46399
1,10365: 1.21656; 1,33817;,46832
1,10720; 1,22333; 1,34797; 1,48110
(d) 1,10603; 1,22110; 1,34473; 1,47688
5. (a) 0,509239; 0,522187;,539023;,559936
0,509701; 0,523155; 0.540550;,562089
0,511127; 0,526155;.545306;,568822
(d) 0,510645; 0,525138:,543690;,566529
6. (a) -0,920498; -0,857538; -0,808030; -0,770038
-0,922575: -0,860923; -0,812300; -0,774965
-0,928059; -0,870054; -0,824021; -0,788686
(d) -0,926341; -0,867163; -0,820279; -0,784275
7. (a) 2 ,90330; 7,53999; 19,4292; 50,5614
(b) 2,93506; 7,70957; 20,1081; 52,9779
5/11/2018 Livro Boyce e Diprima 9th Edicao - Respostas - slidepdf.com
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RESPOSTAS DOS PROBLEMS 58 7
3,03951; 8,28137;2,4562;1,5496
3,00306; 8,07933;1,6163;8,4462
8. (a) 0 ,891830;.25225;,37818;,07257
0,908902;,26872;.39336;,08799
0.958565;,31786;.43924;,13474
(d) 0,942261;.30153;,42389;,11908
9. (a) 3,95713; 5,09853:.41548;.90174
3,95965: 5,10371;,42343;,91255
3,96727; 5,11932; 6,44737;,94512
(d) 3,96473; 5,11411;,43937:,93424
10. (a) 1,60729; 2,46830;.72167;,45963
1.60996: 2.47460:33356;,47774
1,61792; 2,49356:,76940:,53223
(d) 1,61528; 2,48723: 3,75742:,51404
11. (a) -1,45865;0,217545: 1,05715; 1,41487
-1,45322;0,180813; 1.05903: 1.41244
-1,43600;0,0681657;,06489: 1.40575
(d) -1,44190;0,105737; 1,06290; 1.40789
12. (a) 0 ,58798 7;.791589;.14743: 1,70973
0.589440;.795758;,15693; 1,72955
0,593901:.808716;,18687; 1,79291
(d) 0,592396;,804319;,17664; 1.77111
1,595; 2.4636
en+1 = [ 2 f p a n ) - 1 1 h 2 ,  en+Il1 + 2 max0 9 .5 i 10(1 )1 ] e„+1 = e 2 , „ 1 1 2 . lei I < 0,012, e 4 1 < 0,022e„,.1=[20(t„)-7„]h2,  e„. 1 11 + 2 max 0 ,. : i 10 (0 I] h2,e„+, = 2e21„,.2, l e i I < 0,024,e 4 1 < 0,045
e „ + 1 =[7„ +i „) + q 5 3 (inflh 219. e„,_=119 - 15/4-1/2(7„)02/4
e n , ' = (1 +0(701 1121/12/4
e„+1 =1'(7„) + 2i;;1 expl-7„0(i„)] -7„ expl-27„0(i„)I1//2/2
22. (a) CO=1 + (1/57r)sen 57 1b) 1.2; 1,0; 1.2
(c) 1,1; 1,1: 1.0: 1,0d) h <1/.:F5()7.08
e „ + 1 =-0"(i„)h2
(a) 1,55;

,34; 3.46; 5.071,20;,39; 1,57; 1,74
1,20:,42; 1,65; 1,90
26. (a) 0

b) 60

c) -92,16

7. 0,224 0 0.225
Seciio 8.2
1. (a) 1,19512; 1,38120;,55909;,72956
1,19515: 1,38125;,55916:.72965
1,19516; 1,38126;,55918;,72967
2. (a) 1 .62283: 1 ,33460:,12820; 0,995445
1,62243: 1,33386;,12718; 0.994215
1.62234; 1,33368;,12693;,993921
3. (a) 1 ,20526; 1 ,42273;,65511:.90570
1,20533; 1,42290;,65542;,90621
1,20534: 1,42294;,65550:,90634
4. (a) 1 ,10483; 1 ,2188 2;,34146:,47263
1,10484; 1,2188 4;,34147:,47262
1,10484; 1,2188 4;.34147;,47262
5. (a) 0,510164;,524126:,542083: 0,564251
0,510168; 0,524135: 0,542100; 0,564277
0,510169; 0,524137; 0,542104; 0,564284
6. (a) -0,924650; -0,864338: -0,816642: -0,780008
-0,924550; -0,864177; -0,816442; -0,779781
-0,924525: -0,864138; -0,816393; -0,779725
7. (a) 2,96719; 7 ,88313;0,8114;5,5106
(b) 2 ,96800; 7,88 755;0,8294; 55,5758
5/11/2018 Livro Boyce e Diprima 9th Edicao - Respostas - slidepdf.com
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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
1 0 .a),61263: 2 ,4 80 97 ; 3 .7 45 56 : 5 ,4 95 95
(b ),61263; 2 .4 80 92 ; 3 .7 45 50 ; 5 .4 95 89
(a) -1.44768: -0,144478:,06004: 1,40960
(b) -1,44765; -0,143690:,06072: 1,40999
(a) 0.5908 97: 0.799950: 1.16653: 1.74969
(b) 0,590906: 0.799988 : 1.16663: 1.74992
e n + i = (38h 3 /3) exp(47„), le„ 4 . 1 1 < 37. 758 8 h 3 em 0 < t <2, le r I <0,00193389
e „ +1 = (2h 3 /3) exp(27„), len+1 1 < 4,92604h 3 cm 0 < t <1, 'e l < 0,000814269
e„ + 1 =(4h 3 /3)exp(27„).en+11 < 9,85207h 3 e m 0 < tlel< 0.00162854h-' 0,0719. /r,02320. h,0811. h 1 ='• 0,11723. 1,19512. 1 ,38 120 . 1.5 59 09, 1,7 29 56 24. 1,62268, 1,33435, 1,12789, 0.995130
25 . 1,20526, 1,42273. 1.65511. 1,90570 26. 1,10485, 1,21886, 1.34149, 1.47264
Seca() 8.3
(a ),19516; 1,38127: