equacoes Diferenciais
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equacoes Diferenciais


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previo de equac¸o\u2dces diferenciais descubra
7. y = ex 8. y = \u22121
x
9. y = 1 10. y = e
x
2
11. y = c1 + c2x+
x2
2
onde c1 e c2 sa\u2dco constantes arbitra´rias.
14. Sim 15. Sim 16. Na\u2dco
17. (a) Demonstra-se por substituic¸a\u2dco directa e conferindo a condic¸a\u2dco inicial.
(b) Demonstra-se em forma semelhante a` alinha anterior, mas e´ preciso ter em conta que\u221a
a2 = |a|.
(c) Em y = 0 verificam-se as condic¸o\u2dces do teorema de Picard, e como podemos ver no
gra´fico existe soluc¸a\u2dco u´nica em cada caso. Nos pontos y = 0 na\u2dco se verifica a condic¸a\u2dco
de continuidade de \u2202f/\u2202y e existe um nu´mero infinito de soluc¸o\u2dces. Finalmente, em
y < 0 na\u2dco se verifica nenhuma das duas condic¸o\u2dces e na\u2dco existem soluc¸o\u2dces.
Cap\u131´tulo 2
1. y = arcsen
\u221a
2
1 + t2
2. y = t2 \u2212 2t+ 3
3. x = 2
{
arctan
[
\u221a
3 tg
(
y
\u221a
3
2
)]
\u2212 y
}
4. ln
\u2223\u2223y2 \u2212 ty + 2t2\u2223\u2223 = c\u2212 2\u221a
7
arctan
(
2y \u2212 t
t
\u221a
7
)
5. x2 + 2xy + 2y2 = 34
6. y2 + ex sin y = c
7. t+ 15 = (t\u2212 y \u2212 7)(c+ 3 ln |t\u2212 y \u2212 7|)
112 Equac¸o\u2dces de derivadas parciais
8. (y + x/2 + 3/2)
2
(y + x+ 2)3
= 0,098
9. y3 + 3y \u2212 x3 + 3x = 2
10. y =
(
t2 \u2212 2 + 7e\u2212t2/2
)2
11. y = c
x2
+
x3
5
12. (x2 + y2 + 1)e\u2212y2 = c
13. x2 + 2x2y \u2212 y2 = c
14. x3 + y3 \u2212 3xy = c
15. y1 =
1
x
\u2212 2x
x2 + 2c
y2 =
1
x
16. y2 = sinx+
2
c cosx\u2212 sinx y2 = sinx
Cap\u131´tulo 3
1. (4813± 39) anos
2. 10 639 084 habitantes
3. 10 746 263 habitantes
4. 590 m
5. (b) k = 1
t(a\u2212 b) ln
\u2223\u2223\u2223\u2223b(a\u2212 x)a(b\u2212 x)
\u2223\u2223\u2223\u2223 ; x = a 1\u2212 exp
[
kt(a\u2212 b)]
1\u2212 (a/b) exp[kt(a\u2212 b)]
(c) k = 1
at
ln
\u2223\u2223\u2223\u2223 bb\u2212 x
\u2223\u2223\u2223\u2223
(d) k = x
at(a\u2212 x)
6. y4 = cx
7. (a) T \u2032 + T = 15 + 10 sin(2pit)
(b) Tee = 15 + 101 + 4pi2
[
sin(2pit)\u2212 2pi cos(2pit)]
(c) Tm\u131´n = 15\u2212 10\u221a
1 + 4pi2
= 13,4 \u25e6C; Tma´x = 15 +
10\u221a
1 + 4pi2
= 16,6 \u25e6C
11.6 Problemas 113
Cap\u131´tulo 4
2. y = 1
x
(c1 sinx+ c2 cosx)
3. y = c1e2x + c2(2x2 + 2x+ 1)
4. y = c1x+ c2(x2 \u2212 1)
5. y = 2e\u2212x \u2212 e\u22122x
6. y = cosh(ax)
7. y =
1
3
e2x sin(3x)
8. y = (x\u2212 1)ex/4
9. y = 3
4
x2 \u2212 x
10. y = sin(2 ln |x|)
x
11. y = x\u2212 1 + (x\u2212 1)4
12. y = 3e4x
13. Na\u2dco existe soluc¸a\u2dco
14. y = c1 + c2ex + c3e2x
15. y = c1
x
+ x3(c2 + c3 ln |x|)
Cap\u131´tulo 5
1. y = c1ex + c2e\u22122x + 3x
2. y = c1ex + c2e\u2212x \u2212 12(x sinx+ cosx)
3. y =
(
c1 + c2x+
x3
6
)
e2x
4. y = c1 + (c2 \u2212 x)e\u2212x
5. y = c1 sin(2x) + c2 cos(2x)\u2212 14 cos(2x) ln
\u2223\u2223\u2223\u2223tg x+ 1tg x\u2212 1
\u2223\u2223\u2223\u2223
6. y = c1x2 +
c2
x2
+
x2
4
ln |x|+ x
4
12
7. yp =
(
1
2
\u2212 x
)
e\u2212x
8. yp =
1\u221a
x
114 Equac¸o\u2dces de derivadas parciais
Cap\u131´tulo 6
1. {yn} = {1, 0,\u22122, 6, . . .} yn = (\u22121)n(2\u2212 2n)
2. {yn} = {1, 1,\u221215, 81, . . .} yn = (\u22123)n
(
1\u2212 4
3
n
)
3. {yn} = {0, 1, 4, 3, . . .} yn = (
\u221a
13)n
3
sin
[
n arctan
(
3
2
)]
4. {yn} = {0, 1, 2, 0, . . .} yn = 2
n
\u221a
3
sin
(npi
3
)
5. yn = e\u2212n(c12n + c23n)
6. {yn} = {1,\u22123, 3,\u22121, 0, 0, . . .} yn =
\uf8f1\uf8f2\uf8f3
6(\u22121)n
n!(3\u2212 n)! 0 \u2264 n \u2264 3
0 3 < n
7. {yn} = {2, 1, 3, 2/3, 5/4, . . .} y2m = 4m+ 22mm! y2m+1 =
2m(m+ 1)!
(2m+ 1)!
8. {yn} = {1, 1, 0,\u22128,\u22128, 0, . . .} y3m = (\u22128)m y3m+1 = (\u22128)m y3m+2 = 0
9. y3m = c13m \u393
(
m+
1
3
)
y3m+1 = c23m \u393
(
m+
2
3
)
y3m+2 = c33mm!
10. (a) Fn+2 \u2212 Fn+1 \u2212 Fn = 0 F0 = F1 = 1
(c) Fn = 1
\u3c6+ 2
[
\u3c6n+2 + (\u22121)n\u3c6\u2212n]
Cap\u131´tulo 7
1. y = c
\u2211\u221e
n=0 x
2n =
c
1\u2212 x2
2. y = c
\u2211\u221e
n=0
xn
n!
\u2212 x2 \u2212 2x\u2212 3 = cex \u2212 x2 \u2212 2x\u2212 3
3. y = c1
\u2211\u221e
n=0
xn
n!
+ c2
\u2211\u221e
n=0
(2x)n
n!
= c1ex + c2e2x
4. y = c1
\u2211\u221e
n=0
x2n
(2n)!
+ c2
\u2211\u221e
n=0
x2n+1
(2n+ 1)!
\u2212 x = c1 coshx+ c2 sinhx\u2212 x
5. y = c1x+ c2
[
1\u2212\u2211\u221en=0 x2n2nn!(2n\u2212 1)
]
6. y = c1
\u2211\u221e
n=0
(\u22121)n3n \u393
(
n+
1
3
)
(3n)!
x3n + c2
\u2211\u221e
n=0
(\u22121)n3n \u393
(
n+
2
3
)
(3n+ 1)!
x3n+1
11.6 Problemas 115
7. y = c1(1 + x) + c2
\u221a
x
8. y = ex(c1 + c2 lnx)
9. y = c1x+ c2(x2 + x lnx)
10. y = c1
x
+
c2
1\u2212 x
11. y =
\u2211\u221e
n=0
(\u22121)n \u393
(
3
4
)
16nn! \u393
(
n+
3
4
)x4n
12. Ln(x) =
\u2211n
m=0
(\u22121)mn!
(n\u2212m)!m!m!x
m
13. (a) H2m(x) =
\u2211m
k=0
(\u22121)m+k(2m)!
(m\u2212 k)!(2k)! (2x)
2k
(b) H2m+1(x) =
\u2211m
k=0
(\u22121)m+k(2m+ 1)!
(m\u2212 k)!(2k + 1)! (2x)
2k+1
Cap\u131´tulo 8
1. y = 1
6
e\u22122t +
4
3
et \u2212 3
2
2. y = 1
2
t2e\u22122t
3. y = 1
45
e4t \u2212 1
20
e\u2212t +
1
36
(37\u2212 6t)et
4. 8y = 7 cos t\u2212 3 sin t+ e2t(cos t+ sin t)
5. y = t
16
(sin(2t)\u2212 2t cos(2t))\u2212 pi
64
sin(2t)
6. y = t2 \u2212 t
7. y =
1
4
(1\u2212 cos(2t)) [u(t)\u2212 u(t\u2212 pi) + u(t\u2212 2pi)\u2212 u(t\u2212 3pi)]
8. y = 1
2
sin[2(t\u2212 pi)]u(t\u2212 pi)\u2212 1
2
sin(2t)
9. y = 1
4
(1\u2212 cos(2t))[u(t)\u2212 u(t\u2212 pi)] + 1
6
(2 sin t\u2212 sin(2t))u(t\u2212 2pi)
10. 2piy = \u2212 [2t\u2212 2pi \u2212 sin(2t)]u(t\u2212 pi) + [4t\u2212 3pi \u2212 2 cos(2t)]u
(
t\u2212 3pi
4
)
\u22122 [2t\u2212 pi + sin(2t)]u
(
t\u2212 pi
2
)
+ [2t\u2212 pi/2 + cos(2t)]u
(
t\u2212 pi
4
)
+
(
1\u2212 pi
2
)
sin(2t)u(t)
116 Equac¸o\u2dces de derivadas parciais
11. teat
12. t
5
5!
13. t\u2212 sin t
14. e2t \u2212 2t\u2212 1
15. 1
2\u3c93
[sin(\u3c9t)\u2212 \u3c9t cos(\u3c9t)]
16. y = 1
k
\u222b t
0 cosh[k(t\u2212 s)]f(s) ds
17. y = ekt
[
1 + (1\u2212 k)t+ \u222b t0 (t\u2212 s)e\u2212ksf(s) ds]
18. y = ate\u2212t
19. y = 1 + t+ et/2
[
1\u221a
3
sin
(\u221a
3t
2
)
\u2212 cos
(\u221a
3t
2
)]
20. y = 1 + cosh t
21. y = 1
2
sin t+ e\u2212t
(
1\u2212 3
2
t
)
Cap\u131´tulo 9
1. yn = y0 (2\u2212 2n) + y1 (2n \u2212 1) + 2n \u2212 n\u2212 1
2. yn = n
3. yn =
(\u22121)n
8
n(n\u2212 1)2n
4. yn = 2n
(
y0 +
1
2\u2212 e\u2212b
)
\u2212 e
\u2212bn
2\u2212 e\u2212b
5. yn =
2n
4
[
1\u2212 cos
(npi
3
)
\u2212 1\u221a
3
sin
(npi
3
)]
6. y2m =
1
37
[
28(\u22124)m + 9
9m
]
y2m+1 = \u2212 337
[
(\u22124)m + 1
9m
]
7. y2m = y0 +m(m\u2212 1) y2m+1 = y1 +m2
8. y(z) = 1
9. y(z) = 1
z(z \u2212 1)
10. y(z) = z(z
2 \u2212 1) sin\u3c9
(z2 \u2212 2z cos\u3c9 + 1)2
11.6 Problemas 117
11. 1. Tn+1 \u2212 Tn = n+ 1 T1 = 1
2. Tn =
n(n+ 1)
2
12. Sn = T 2n =
n2(n+ 1)2
4
13. Sn = n(n+ 1)
14. O per\u131´odo e´ 4, 3 e 1 respectivamente. Existem pontos de bifurcac¸a\u2dco entre \u22122 e \u22121.75, e
entre \u22121.75 e \u22121.3
Cap\u131´tulo 10
1. x = sin t y = cos t+ sin t
2. x = 1
2
(sin t\u2212 t cos t) y = 1
2
(sin t\u2212 t cos t+ t sin t)
3.
\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3
x =
2e\u2212t/2\u221a
3
sin
(\u221a
3t
2
)
y = \u2212e\u2212t/2
[
cos
(\u221a
3t
2
)
+
1\u221a
3
sin
(\u221a
3t
2
)]
z = e\u2212t/2
[
cos
(\u221a
3t
2
)
\u2212 1\u221a
3
sin
(\u221a
3t
2
)]
4. x =
[
2e2t \u2212 et
et
]
5. x =
\uf8ee\uf8f0 e2t \u2212 e3te3t
\u2212e2t + e3t
\uf8f9\uf8fb
6. x =
[
cos(2t) + 7 sin(2t)
2 cos(2t)\u2212 6 sin(2t)
]
7. x = et
\uf8ee\uf8ef\uf8ef\uf8f0
1
\u22121 + 2 cos(2t) + 1
2
sin(2t)
3
2
\u2212 1
2
cos(2t) + 2 sin(2t)
\uf8f9\uf8fa\uf8fa\uf8fb
8. x = e2t
[
1 + t
1
]
9. x = e\u2212t
\uf8ee\uf8f0 \u2212t1
e\u2212t
\uf8f9\uf8fb
118 Equac¸o\u2dces de derivadas parciais
10. x = 1
5
\uf8ee\uf8f0 2e\u221210t + 3e5t5e5t
\u2212e\u221210t + 6e5t
\uf8f9\uf8fb
11. x = et
\uf8ee\uf8ef\uf8ef\uf8f0
cos(4t)\u2212 sin(4t)
sin(4t) + cos(4t)
2et \u2212 1
1
\uf8f9\uf8fa\uf8fa\uf8fb
12. x = 1
48
[ \u22123\u2212 12t+ 16et + 35e4t
\u22123\u2212 12t\u2212 32et + 35e4t
]
13. x = et
\uf8ee\uf8ef\uf8ef\uf8ef\uf8f0
t+
t2
2
1 + t
t2
2
\uf8f9\uf8fa\uf8fa\uf8fa\uf8fb
14. x =
[
sin(2t) + cos(2t)\u2212 u(t\u2212 pi) sin(2t)
2 sin(2t) + u(t\u2212 pi)[cos(2t)\u2212 sin(2t)]
]
15. x = te
t
6
\uf8ee\uf8f0 6t\u2212 t23t
6 + 6t+ t2
\uf8f9\uf8fb
16. x = 1
2
[
3\u2212 4e\u2212t + e\u22122t \u2212 u(t\u2212 1)(3\u2212 4e1\u2212t + e2\u22122t)
2\u2212 4e\u2212t + 2e\u22122t \u2212 u(t\u2212 1)(2\u2212 4e1\u2212t + 2e2\u22122t)
]
Cap\u131´tulo 11
1. u(x, y) = xy + f(y), onde f e´ qualquer func¸a\u2dco de y deriva´vel
2. u(x, y) = f(x) + g(y), onde f e g sa\u2dco func¸o\u2dces de x e y, ambas deriva´veis nas respectivas
varia´veis
3. u(x, y) = 1
3
x3y +
1
3
xy3 + f(x) + g(y), onde f e g sa\u2dco func¸o\u2dces de x e y, ambas deriva´veis
nas respectivas varia´veis
4. v(x, t) = 2e\u2212x/2
(
t\u2212 x
2
) [
u
(
t\u2212 x
2
)
\u2212 u
(
t\u2212 1\u2212 x
2
)]
5. v(x, t) = sin
(
t\u2212 x
c
)
u
(
t\u2212 x
c
)
6. u(x, t) = x
(
t\u2212 1 + e\u2212t)
7. f(x) =
4
pi
\u2211\u221e
n=1