Lista_9

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Universidade de Brasília
Departamento de Matemática
Cálculo 2
Lista de Exercícios - Semana 09
1. Resolva as seguintes equações diferenciais:
(a) y′′ − y′ − 30y = 0
(b) y′′ − 2y′ + y = 0
(c) y′′ − 7y = 0
(d) y′′ + 2y′ + 2y = 0
(e) y′′ + 6y′ + 9y = 0
(f) y′′ + 2y′ + 3y = 0
(g) y′′ − 3y′ − 5y = 0
(h) 4y′′ + 4y′ + y = 0
(i) y′′ + y′ + 2y = 0
(j) y′′′ − 2y′′ − y′ + 2y = 0
(k) y′′′ − y′′ − y′ + y = 0
(l) y′′′ − 3y′′ + 3y′ − y = 0
(m) y′′′ − y′′ + y′ − y = 0
(n) y(4) + 2y′′ + y = 0
2. Resolva as seguintes equações diferenciais:
(a) y′′ − 10y′ + 25y = e5x
(b) y′′′ + y = x
(c) y′′ + 49y = cos 7x
(d) y′′ + 4y′ + 8y = e2x(sen2x+ cos 2x)
(e) y′′ + y′ = senx+ cosx
(f) y′′ − 4y′ = cos2 x
(g) y(4) + 2y′′′ + y′′ = e4x
(h) y(4) − y′′′ = 6
GABARITO
1. (a) y = c1e6x + c2e−5x
(b) y = c1ex + c2xex
(c) y = c1e
√
7x + c2e
−
√
7x
(d) y = c1e−x cosx+ c2e−xsenx
(e) y = c1e−3x + c2xe−3x
(f) y = c1e−x cos
√
2x+ c2e
−xsen
√
2x
(g) y = c1e
3+
√
29
2
x + c2e
3−
√
29
2
x
(h) y = c1e−
x
2 + c2xe
−x
2
(i) y = c1e−x/2 cos
√
7
2 x+ c2e
−x/2sen
√
7
2 x
(j) y = c1e2x + c2e−x + c3ex
(k) y = c1e−x + c2ex + c3xex
(l) y = c1ex + c2xex + c3x2ex
(m) y = c1ex + c2 cosx+ c3senx
(n) y = c1 cosx+ c2x cosx+ c3senx+ c4xsenx
2. (a) y = c1e5x + c2x5x + x
2e5x
2
(b) y = c1e−x + c2ex/2 cos
(√
3x
2
)
+ c3e
x/2sen
(√
3x
2
)
+ x
(c) y = c1sen7x+ c2 cos 7x+ 7xsen7x98
(d) y = c1e−2xsen2x+ c2e−2x cos 2x+ e
2xsen2x
16
(e) y = c1 + c2e−x − cosx
(f) y = c1 + c2e4x − 1160 (8sen2x+ 4 cos 2x+ 20x)
(g) y = c1 + c2x+ c3e−x + c4xe−x + e
4x
400
(h) y = c1 + c2x+ c3x2 + c4ex − x3