Lista 2 prof. Walter MAT 147 2016 I

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Segunda Lista de Exerc´ıcios de MAT147
Em cada um dos seguintes exerc´ıcios, ache a se´rie de poteˆncias para a func¸a˜o dada e apresente
o raio de convergeˆncia do mesmo.
1. Ln(x+ 1) em poteˆncias de (x− 1)
Rpta:
∞∑
n=1
(−1)n−1 (x− 1)
n
n
, r = 1
2.
√
x em poteˆncias de x− 4
Rpta: 2 + 1
4
(x− 4) + 2
∞∑
n=2
(−1)n−11 · 3 · 5 · · · (2n− 3)(x− 4)
n
2 · 4 · 6 · · · (2n)4n , r =
3. Ln|x| em poteˆncias de x+ 1
Rpta: −
∞∑
n=
(x+ 1)n
n
, r = 1
4. 1−cos(x)
x
em poteˆncias de x
Rpta:
∞∑
n=1
(−1)n+1x
2n−1
(2n)!
, r = +∞
5. 4x4 − 15x3 + 20x2 − 10x+ 14 em poteˆncias de x+ 1
Rpta: 63− 111(x+ 1) + 89(x+ 1)2 − 31(x+ 1)3 + 4(x+ 1)4, r =
6. sen2(x) em poteˆncias de x
Rpta: 1
2
∞∑
n=1
(−1)n−1 (2x)
2n
(2n)!
, r =
7. 2x em poteˆncias de x
8. 3
(1−x)(1+2x) em poteˆncias de x
Rpta:
∞∑
n=
(1 + (−1)n2n+1)xn, r =
Nos seguintes exerc´ıcios encontre o intervalo de convergeˆncia da se´rie de poteˆncia dada.
1.
+∞∑
n=0
(−1)n+1 x
2n−1
(2n− 1)!
Rpta: I = R
2.
+∞∑
n=0
2nx2
n2
Rpta: I = [−1
2
; 1
2
]
3.
+∞∑
n=0
n!xn
Rpta: I = {0}
1
4.
+∞∑
n=0
(−1)n x
n
(2n− 1)32n−1
Rpta: I = (−9; 9]
5.
+∞∑
n=0
(−1)n+n (x− 1)
n
n
Rpta: I = (0; 2]
6.
+∞∑
n=0
n!xn
nn
Rpta: I = (−e; e)
7.
+∞∑
n=1
nn(x− 3)n
Rpta: I =
8.
+∞∑
n=1
(
n
2n+ 1
)2n−1xn
Rpta: I = (−4; 4)
9.
+∞∑
n=1
(3x− 1)n
n2 + n
Rpta: I = [0; 2
3
]
10.
+∞∑
n=2
xn−1
n3nLn(n)
Rpta: I = (−2, 2)
11.
+∞∑
n=0
3n
2
xn
2
Rpta: I = (−1
3
; 1
3
)
Resolver as seguintes EDO’s (Varia´veis Separa´veis):
1.
tg(x)sen2(y)dx+ cos2(x)ctg(y)dy = 0
Rpta: ctg2(y) = tg2(x) + c
2.
xy
′ − y = y3
Rpta: x = cy√
1+y2
2
3. √
1 + x3
dy
dx
= x2y + x2
Rpta: 2
√
1 + x2 = 3Ln(y + 1) + c
4.
e2x−ydx+ ey−2xdy = 0
Rpta: e4x + 2e2y = c
5.
(x2y − x2 + y − 1)dx+ (xy + 2x− 3y − 6)dy = 0
Rpta: x
2
2
+ 3x+ y + Ln(x− 3)10(y − 1)3 = c
6.
ex+ysen(x)dx+ (2y + 1)e−y
2
dy = 0
Rpta: ex(sen(x)− cos(x))− 2e−y2−y = 0
7.
ey(
dy
dx
+ 1) = 1
Rpta: Ln(ey − 1) = C − x
8.
y
′
= 1 + x+ y2 + xy2
Rpta: arctg(y)− x− x2
2
= c
9.
y − xy′ = a(1 + x2y)
Rpta: y = a+cx
1+ax
10.
e−y(1 + y
′
) = 1
Rpta: ex = C(1− e−y)
11.
y
′
= 10x+y
a > 0, a 6= 1
Rpta: 10x + 10−y = C
12.
dy
dx
=
x2
y(1 + x3)
Rpta: 3y2 − 2Ln(1 + x3) = C
3
13.
dy
dx
=
x− e−x
y + ey
Rpta: y2 − x2 + 2(ey − e−x) = C
14.
dy
dx
=
ax+ b
cx+ d
a, b, c, d ∈ R
Rpta: y = ax
c
+ bc−ad
c2
Ln(|cx+ d|) +K
Nos seguintes exercicios, achar a soluc¸a˜o particular da EDO, com a restric¸a˜o dada (Varia´veis
Separa´veis):
1.
y
′ − 2yctg(x) = 0, y(pi
2
) = 2
Rpta: y = 2sen2(x)
2.
dy
dx
=
x
y
− x
1 + y
, y(0) = 1
Rpta: 3y2 + 2y3 = 3x2 + 5
3.
x(y6 + 1)dx+ y2(x4 + 1)dy = 0, y(0) = 1
Rpta: 3arctg(x2) + 2arctg(y3) = pi
2
4. √
1− cos(2x)
1 + sen(y)
+ y
′
= 0, y(
pi
4
) = 0
Rpta:
√
2sen(x) + sen(y)− cos(y) = 0
5.
y2y
′ − x2 = 0, y(−2) = −2
Rpta: y = x
6.
x3dy + xydx = x2dy + 2ydx, y(2) = e
Rpta: xy = 2(x− 1)e 2x
7.
dy
dx
= xy3(1 + x2)
−1
2 , y(0) = 1
Rpta: y = (3− 2√1 + x2)−12
4
8.
y
′
sen(x) = yLn(y), y(
pi
2
) = e
Rpta: y = etg(
x
2
)
9.
(1 + ex)yy
′
= ex, y(0) = 1
Rpta: .
10.
(xy2 + x)dx+ (x2y − y)dy = 0, y(0) = 1
Rpta: 1 + y2 = 2
1−x2
11.
(4x+ xy2)dx+ (y + x2y)dy = 0, y(1) = 2
Rpta: (1 + x2)(4 + y2) = 16
12.
xdx+ ye−xdy, y(0) = 1
Rpta: y = [2(1− x)ex − 1] 12
13.
yey
2
y
′
= x− 1, y(2) = 0
Rpta: ey
2
= x2 − 2x+ 1
14.
y
′
+ 6ytg(2x) = 0, y(0) = −2
Rpta: y = −2cos3(2x)
Resolver as seguintes EDO’s (Reduc´ıveis a Varia´veis Separa´veis):
1.
(x6 − 2x5 + 2x4 − y3 + 4x2y)dx+ (xy2 − 4x3)dy = 0
Sug: y = xz Rpta: x
3
3
− x2 + 2x+ y3
3x3
− 4y
x
= c
2.
y
′
=
y − x+ 1
y − x+ 5
Rpta: (y − x)2 + 10x− 2x = c
3.
ye
x
y2 dx+ (y2 − 2xe xy2 )dy = 0
Rpta: Ln(y) + e
x
y2 = C
5
4.
y
′
= sen(x− y)
Rpta: x+ c = ctg(y−x
2
− pi
4
)
5.
y
′
= (8x+ 2y + 1)2
Rpta: 8x+ 2y + 1 = 2tg(4x+ c)
6.
(1− xycos(xy))dx− x2cos(xy)dy = 0
Rpta: Ln(x)− sen(xy) = C
7.
(x2y3 + y + x− 2)dx+ (x3y2 + x)dy = 0
Rpta: 3x2 − 12x+ 2x3y3 + 6xy = C
8.
[x2sen(
y
x2
− 2ycos( y
x2
))]dx+ xcos(
y
x2
)dy = 0
Rpta: xsen( y
x2
) = C
9.
eyy
′
= K(x+ ey)− 1
Sug: z = x+ ey Rpta: y = Ln(CeKx−x)
10.
x2yy
′
=
1
2
tg(x2y2)− xy2
Sug: z = x2y2 Rpta: sen(x2y2) = kex
11.
(x2y2 + 1)dx+ 2x2dy = 0
Rpta: 1
1−xy +
1
2
Ln(x) = C
Resolva as seguintes EDO’s Lineares:
1.
(2xy
′
+ y)
√
1 + x = 1 + 2x
Rpta: y = c√
x
+
√
1 + x
2.
x(1− x2)dy
dx
− y + ax3 = 0
Rpta: y = ax+ cx√
1−x2
6
3.
(y2 − 1)dx = y(x+ y)dy
Rpta: x =
√
y2 − 1(Ln(y +√y2 − 1) + c)− y
4.
(1 + y2)dx = (
√
1 + y2sen(y)− xy)dy
Rpta: x
√
1 + y2 + cos(y) = c
5.
dy
dx
+ ycos(x) = sen(x)cos(X)
Rpta: Y = Ce−sen(x) + sen(x)− 1
6.
dy
dx
(xcos(y) + asen(2y)) = 1
Rpta: x = cesen(y) − 2a(1 + sen(y))
7.
dy
dx
+ y = sen(x)
Rpta: y = ce−x + 1
2
(sen(x)− cos(x))
8.
dy
dx
+ xy = 2x
Rpta: y = ce
x2
2 + 2
9.
x2dy − sen(2x)dx+ 3xydx = 0
Rpta: 4x3y + 2xcos(2x) = csen(2x)
10.
dy
dx
=
3 + xy
2x2
Rpta: y = c
√
x− 1
x
11.
x
dy
dx
+
y
1 + x
= 1− x2
Rpta: y = 1 + x
2
− x2
2
+ c(1 + 1
x
)
12.
(x+ y)2(xdy − ydx) + [y2 − 2x2(x+ y)2](dx+ dy) = 0
Rpta: (y − x2 − xy)(x+ y)3 = k(y) + 2x2 + 2xy
Resolver as seguintes EDO’s (Bernoulli):
7
1.
dy
dx
+
1
x− 2y = 5(x− 2)
√
y
Rpta: y
1
2 = c(x− 2)−12 + (x− 2)2
2.
dy
dx
= − y
3
e2x + y2
Sug: z = e2x Rpta: y2 = (c− 2Ln(|y|))e2x
3.
2cos(y)dx− (xsen(y)− x3)dy = 0
Rpta: sen(y) = x2(c+ tg(y))
4.
dy +
1
x
ydx = 3x2y2dx
Rpta: xy(c− 3x2
2
) = 1
5.
dy + ydx = 2xy2exdx
Rpta: 1 = yex(c− x2)
6.
3
dy
dx
+
3
x
y = 2x4y4
Rpta: x−3y−3 + x2 = c
7.
dx
x
= (xsen(y)− 1)dy
Rpta: 1
x
= cey + 1
2
(sen(y) + cos(y))
8.
dy
dx
=
3x2
x3 + y + 1
Rpta: x3 = cey − y − 2
9.
8xy
′ − y = − 1
y3
√
x+ 1
Rpta: y4 = x
√
x+
√
x+ 1
10.
x
dy
dx
+
y
Ln(x)
=
x(x+ Ln(x))
y2Ln(x)
Rpta: y = [3x+ 3
2
(x
2−6x
Ln(x)
)− 3
2
(x
2−12x
Ln(x2)
) + 1
4
(3x
2−72x+c
(Ln(x))3
)]
1
3
8
Resolver as seguintes EDO’s homogeˆneas de coeficientes constantes:
1.
d2y
− 4dx
2 dy
dx
+ 4y = 0
Rpta: y = e2x(c1x+ c2)
2.
d2y
− 3dx
2 dy
dx
+ 2y = 0
Rpta: y = c1e
x + c2e
2x
3.
d2y
dx2
+ y = 0
Rpta: y = c1cos(x) + c2sen(x)
4.
d2y
dx2
+
dy
dx
+ y = 0
Rpta: y = e
x
2 [c1cos(
√
3
2
x) + c2sen(
√
3
2
x)]
5.
d2y
dx2
+ 2
dy
dx
+ 2y = 0
Rpta: y = e−x(c1cos(x) + c2sen(x))
6.
d2y
dx2
+ k2y = 0
Rpta: y = Acos(kx) +Bsen(kx)
7.
2
d2y
dx2
− 3dy
dx
+ y = 0
Rpta: y = c1e
x
2 + c2e
x
8.
d2y
dx2
− 9dy
dx
+ 9y = 0
Rpta: y = Ae
(9+3
√
5)x
2 +Be
(9−3√5)x
2
9.
d2y
dx2
+
dy
dx
− 2y = 0
y(0) = 1 e y
′
(0) = 1
Rpta: y = ex
9
10.
d2y
dx2
− 6dy
dx
+ 9y = 0
y(0) = 0 e y
′
(0) = 2 Rpta: y = 2xe3x
11.
d2y
dx2
+ 8
dy
dx
− 9y = 0
y(1) = 1 e y
′
(1) = 0 Rpta: y = 1
10
e−9(x−1) + 9
10
ex−1
12.
d2y
dx2
+ 4
dy
dx
5y = 0
y(0) = 1 e y
′
(0) = 0 Rpta: y = e−2xcos(x) + 2e−2xsen(x)
13.
d2y
dx2
− 6dy
dx
+ 25y = 0
Rpta: y = e3x(Acos(4x) +Bsen(4x))
14.
d2y
dx2
− 12dy
dx
+ 35y = 0
Rpta: y = Ae5x +Be7x
15.
9
d2y
dx2
− 30dy
dx
+ 25y = 0
Rpta: y = (A+Bx)e
5x
3
16.4
d2y
dx2
− 20dy
dx
+ 25y = 0
Rpta: y = (A+Bx)e2.5x
10