lista 12_calc 02A 2011
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lista 12_calc 02A 2011


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Lista 12 Ca´lculo II - A 2010-2 27
Universidade Federal Fluminense
EGM - Instituto de Matema´tica
GMA - Departamento de Matema´tica Aplicada
LISTA 12 - 2010-2
EDO linear de ordem n
com coeficientes constantes:
Me´todo dos coeficientes indeterminados
Me´todo de variac¸a\u2dco dos para\u2c6metros
Nos exerc´\u131cios 1 a 12 encontre a soluc¸a\u2dco geral da EDO linear homoge\u2c6nea.
1. y\u2032\u2032 \u2212 36y = 0
2. y\u2032\u2032 + 9y = 0
3. y\u2032\u2032 \u2212 y\u2032 \u2212 6y = 0
4.
d2y
dx2
+ 8
dy
dx
+ 16y = 0
5. y\u2032\u2032 + 3y\u2032 \u2212 5y = 0
6. y\u2032\u2032 \u2212 4y\u2032 + 5y = 0
7. 3y\u2032\u2032 + 2y\u2032 + y = 0
8. y\u2032\u2032\u2032 \u2212 y = 0
9. y\u2032\u2032\u2032 \u2212 5y\u2032\u2032 + 3y\u2032 + 9y = 0
10. y\u2032\u2032 + y\u2032\u2032 \u2212 2y = 0
11. 16yiv + 24y\u2032\u2032 + 9y = 0
12. y(5) \u2212 16y\u2032 = 0
Nos exerc´\u131cios 13 e 14 resolva o PVI.
13. y\u2032\u2032\u2032 + 12y\u2032\u2032 + 36y\u2032 = 0; y(0) = 0; y\u2032(0) = 1; y\u2032\u2032(0) = \u22127
14. y(4) \u2212 3y(3) + 3y\u2032\u2032 \u2212 y\u2032 = 0; y(0) = y\u2032(0) = 0; y\u2032\u2032(0) = y\u2032\u2032\u2032(0) = 1
Nos exerc´\u131cios 15 a 21 resolva as equac¸o\u2dces, usando o me´todo dos coeficentes indeterminados.
15. y\u2032\u2032 \u2212 y\u2032 + 1
4
y = 3 + ex/2
16. y\u2032\u2032 + y = 2x senx
17. y\u2032\u2032 + 4y = (x2 \u2212 3) sen 2x
18. y\u2032\u2032 + 2y\u2032 + y = senx+ 3 cos 2x
19. y\u2032\u2032\u2032 \u2212 3y\u2032\u2032 + 3y\u2032 \u2212 y = x\u2212 4ex
20. y\u2032\u2032\u2032 \u2212 y\u2032\u2032 + y\u2032 \u2212 y = xex \u2212 e\u2212x + 7
21. 16y(4) \u2212 y = ex/2
Nos exerc´\u131cios 22 a 24 resolva as equac¸o\u2dces.
22. y\u2032\u2032 \u2212 y = 1/x, x > 0
23. 4y\u2032\u2032 + 36y = csc 3x , x \u2208 (0, pi/6)
24. y\u2032\u2032\u2032 \u2212 y\u2032\u2032 + y\u2032 \u2212 y = e\u2212x senx
Nos exerc´\u131cios 25 e 26 resolva o PVI.
25. yiv + 2y\u2032\u2032 + y = senx, y(0) = 2; y\u2032(0) = 0; y\u2032\u2032(0) = \u22121; y\u2032\u2032\u2032(0) = 1
26. y\u2032\u2032\u2032 \u2212 y\u2032\u2032 + y\u2032 \u2212 y = secx; y(0) = 2; y\u2032(0) = \u22121; y\u2032\u2032(0) = 1
Lista 12 Ca´lculo II - A 2010-2 28
RESPOSTAS DA LISTA 12 (Com indicac¸a\u2dco ou resumo de algumas resoluc¸o\u2dces)
1. y(x) = C1e6x + C2e\u22126x
2. y(x) = C1 cos 3x+ C2 sen 3x
3. y(x) = C1e3x + C2e\u22122x
4. y(x) = C1e\u22124x + C2 xe\u22124x
5. y(x) = C1e
\u22123\u2212\u221a29
2
x + C2e
\u22123+\u221a29
2
x
6. y(x) = C1e2x cosx+ C2e2x senx
7. y(x) = C1e
\u2212x
3 cos
\u221a
2x
3 + C2e
\u2212x
3 sen
\u221a
2x
3
8. y(x) = C1ex + C2e
\u2212x
2 cos
\u221a
3x
2 + C3e
\u2212x
2 sen
\u221a
3x
2
9. y(x) = C1e\u2212x + C2e3x + C3 xe3x
10. y(x) = C1ex + e\u2212x (C2 cosx+ C3 senx)
11. y(x) = C1 cos
(\u221a
3x
2
)
+ C2 sen
(\u221a
3x
2
)
+ C3 x cos
(\u221a
3x
2
)
+ C4 x sen
(\u221a
3x
2
)
12. y(x) = C1 + C2e2x + C3e\u22122x + C4 cos 2x+ C5 sen 2x
13. y(x) = 536 \u2212 536e\u22126x + x6e\u22126x
14. y(x) = 23 \u2212 23ex + 23 xex \u2212 16 x2ex
15. y(x) = C1ex/2 + C2 xex/2 + 12 +
x2ex/2
2
16. y(x) = C1 cosx+ C2 senx\u2212 x
2 cosx
2
+
x senx
2
17. y(x) = C1 cos 2x+ C2 sen 2x+ 2532x cos 2x+
1
16x
2 sen 2x\u2212 112x3 cos 2x
18. y(x) = C1e\u2212x + C2 xe\u2212x \u2212 cosx2 +
12 sen 2x
25
\u2212 9 cos 2x
25
19. y(x) = C1ex + C2 xex + C3 x2ex \u2212 x\u2212 3\u2212 23x3ex
20. y(x) = C1 + C2 cosx+ C3 senx\u2212 7 + 14 e\u2212x \u2212 12 xex + 14x2ex
21. y(x) = C1ex/2 + C2e\u2212x/2 + C3 cos (x/2) + C4 ( senx/2) + 18 e
x/2
22. y = C1ex + C2e\u2212x + 12e
x
\u222b x
x0
e\u2212t
t
dt\u2212 1
2
e\u2212x
\u222b x
x0
et
t
dt
23. y = C1 cos 3x+ C2 sen 3x\u2212 x cos 3x12 +
sen 3x
36
ln | sen 3x|, x \u2208 (0, pi/6)
24. y = C1ex + C2 cosx+ C3 senx\u2212 12x2 senx
25. y = 2 cosx+ 78 senx\u2212 78 x cosx+ 12 x senx\u2212 18x2 senx
26. y = 32 +
1
2 cosx\u2212 52 senx\u2212 12(cosx) ln(cosx) +\u221212( senx) ln(cosx)\u2212 12x cosx\u2212
\u221212 senx+ 12ex
\u222b x
0
e\u2212x
cosx
dx