Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. -3 -2 х, 3 -[-] 5. х x(0)
Find a fundamental matrix of each of the systems in Problems 1 through 8, then apply Eq. (8) to find a solution satisfying the given initial conditions. -3 -2 х, 3 -[-] 5. х x(0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Find a fundamental matrix of each of the systems in Problems
1 through 8, then apply Eq. (8) to find a solution satisfying the
given initial conditions.
-3
-2
х,
3
|
5. x'
x(0) = |
9.
3
2
8. x' =
-5
-4
-2
X,
x(0)
5
3
THEOREM 1
Fundamental Matrix Solutions
Let (1) be a fundamental matrix for the homogeneous linear system x' = Ax.
Then the [unique] solution of the initial value problem
x' = Ax,
x(0) = Xo
(7)
is given by
x(1) = (1)(0)-Xo.
(8)
I found this answer in the book
2 cos 3t
-2 sin 3t
5. $(t) =
-3 cos 3t +3 sin 3t
3 cos 3t +3 sin 3t
3 cos 31 – sin 3t
x(t) =
%3D
-3 cos 3t + 6 sin 3t
[
e3
-e3
el
8. Ф(()
e-21
-e
x(t)
-e' +e-2
%3D
2r
e-2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdff89607-66f5-4cb4-8e83-4a60bfa7ae5f%2F68d942ed-d936-42ae-8159-3ec591962bea%2Fkyo18a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find a fundamental matrix of each of the systems in Problems
1 through 8, then apply Eq. (8) to find a solution satisfying the
given initial conditions.
-3
-2
х,
3
|
5. x'
x(0) = |
9.
3
2
8. x' =
-5
-4
-2
X,
x(0)
5
3
THEOREM 1
Fundamental Matrix Solutions
Let (1) be a fundamental matrix for the homogeneous linear system x' = Ax.
Then the [unique] solution of the initial value problem
x' = Ax,
x(0) = Xo
(7)
is given by
x(1) = (1)(0)-Xo.
(8)
I found this answer in the book
2 cos 3t
-2 sin 3t
5. $(t) =
-3 cos 3t +3 sin 3t
3 cos 3t +3 sin 3t
3 cos 31 – sin 3t
x(t) =
%3D
-3 cos 3t + 6 sin 3t
[
e3
-e3
el
8. Ф(()
e-21
-e
x(t)
-e' +e-2
%3D
2r
e-2
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