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UTFPR - Universidade Tecnolo´gica Federal do Parana´
Pato Branco
Engenharias
Lista de Exerc´ıcios
Sistemas de equac¸o˜es diferenciais lineares-VARIAC¸A˜O DOS PARAˆMETROS
1- Aplique a variac¸a˜o dos paraˆmetros para resolver o sistema dado.
a)

dx
dt
= 3x− 3y + 4
dy
dt
= 2x− 2y − 1
b) X ′ =
(
3 −5
3
4
−1
)
X +
(
1
−1
)
e
t
2
c) X ′ =
(
0 2
−1 3
)
X +
(
1
−1
)
et
d) X ′ =
(
1 8
1 −1
)
X +
(
12
12
)
t
e) X ′ =
(
3 2
−2 −1
)
X +
(
2e−t
e−t
)
f) X ′ =
(
0 −1
1 0
)
X +
(
sec t
0
)
g) X ′ =
(
1 −1
1 1
)
X +
(
cos t
sin t
)
et
h) X ′ =
(
0 1
−1 0
)
X +
(
0
sec t tan t
)
Gilson Tumelero 2
i) X ′ =
(
1 2
−1
2
1
)
X +
(
csc t
sec t
)
et
j) X ′ =
 1 1 01 1 0
0 0 3
X +
 ete2t
te3t

k) X ′ =
(
2 −1
3 −2
)
X +
(
et
t
)
l) X ′ =
(
2 −5
1 −2
)
X +
( − cos t
sin t
)
m) X ′ =
(
1 1
4 −2
)
X +
(
e−2t
−2et
)
n) X ′ =
(
4 −2
8 −4
)
X +
(
t−3
−t−2
)
, t > 0
o) X ′ =
( −4 2
2 −1
)
X +
(
t−1
2t−1 + 4
)
, t > 0
p) X ′ =
(
1 1
4 1
)
X +
(
2
−1
)
et
q) X ′ =
(
2 −1
3 −2
)
X +
(
1
−1
)
et
r) X ′ =
( −3 √2√
2 −2
)
X +
(
e−t
−e−t
)
s) X ′ =
(
2 −5
1 −2
)
X +
(
0
cos t
)
, 0 < t < Π
t) X ′ =
(
2 −5
1 −2
)
X +
(
csc t
sec t
)
, Π
2
< t < Π
Gilson Tumelero 3
2- Resolva o sistema dado sujeito a` condic¸a˜o inicial indicada.
a) X ′ =
(
3 −1
−1 3
)
X +
(
4e2t
4e4t
)
, X(0) =
(
1
1
)
b) X ′ =
(
4 1
6 5
)
X +
(
50e7t
0
)
, X(0) =
(
5
−5
)