Lista5 - TurmaT01

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UNIVERSIDADE FEDERAL DE SERGIPE
DEPARTAMENTO DE MATEMA´TICA
Disciplina: Equac¸o˜es Diferenciais Ordina´rias Turmas: T01
Prof.: Adriano Veiga de Oliveira Data: 29/08/19
QUINTA LISTA DE EXERCI´CIOS
1. Encontre a soluc¸a˜o geral do sistema de equac¸o˜es diferenciais homogeˆneo de primeira ordem
dado.
(a)
{
x′ = x + y
y′ = 4x + y
(b) ~X ′ =
( −1
2
1
−1 −1
2
)
~X (c)
{
x′ = x + 9y
y′ = −x− 5y
(d) ~X ′ =
 0 1 11 0 1
1 1 0
 ~X (e)

x′ = 2x + y
y′ = x + 4y + z
z′ = 2x− 2y + 3z
(f)

x′ = 2x + y + 2z
y′ = −y + z
z′ = −y − z
2. Use a transformada de Laplace para resolver o problema de valor inicial{
x′′ = −2x′ − 5y + 3
y′ = x′ + 2y
, x(0) = 0, x′(0) = 0, y(0) = 1
3. Encontre a soluc¸a˜o geral do sistema de equac¸o˜es diferenciais de primeira ordem na˜o homogeˆneo
dado.
(a)
{
x′ = x + 2y + 6e3t
y′ = 2x + y + 2e3t
(b) ~X ′ =
( −3 2
−4 1
)
~X +
(
0
2 sin t
)
(c) ~X ′ =
 1 1 01 1 0
0 0 3
 ~X +
 ete2t
te3t

QUINTA LISTA - GABARITO
1. (a)
(
x
y
)
= c1
(
1
−2
)
e−t + c2
(
1
2
)
e3t
(b)
{
x = c1e
−t/2 cos t + c2e−t/2 sin t
y = −c1e−t/2 sin t + c2e−t/2 cos t
(c)
(
x
y
)
= c1
(
3
−1
)
e−2t + c2
[( 3
−1
)
t +
(
1
0
)]
e−2t
(d)
 xy
z
 = c1
 11
1
 e2t + c2
 10
−1
 e−t + c3
 01
−1
 e−t
(e)
 xy
z
 =
 11
−2
 (c1 + c2t + c3 t22 )e3t +
 01
0
 (c2 + c3t)e3t + c3
 00
1
 e3t
(f)

x = c1e
2t + c2e
−t(− cos t + 7 sin t) + c3e−t(−7 cos t− sin t)
y = c2e
−t(10 cos t) + c3e−t(10 sin t)
z = c2e
−t(−10 sin t) + c3e−t(10 cos t)
2.
{
x = 2 cos t + 6 sin t− 2− 6t
y = −2 cos t + 2 sin t + 3
3. (a)
(
x
y
)
= c1
(
1
1
)
e3t + c2
( −1
1
)
e−t +
(
4
4
)
te3t +
1
2
(
1
−1
)
e3t
(b)
{
x = c1e
−t cos 2t + c2e−t sin t− 25 cos t + 45 sin t
y = −c1e−t(cos 2t− sin 2t) + c2e−t(cos 2t + sin 2t)− 15 cos t + 75 sin t
(c)
 xy
z
 = c1
 1−1
0
+ c2
 11
0
 e2t + c3
 00
1
 e3t +
 −14e2t + 12te2t−et + 1
4
e2t + 1
2
te2t
1
2
t2e3t
