1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. (a) x' = (c) x' = 5 2 4 2 X; 0 -1 -2 -4 -2 x; (b) x' = (d) x' 1 X, 9 -6 6 x(0) = []; 4 -1 ; Sol'n 0 x, x(0) 4 3 -4. =
1. Use Theorem 1 to obtain the general solution of the given first-order systems; if initial conditions are given, find the particular solution satisfying them. (a) x' = (c) x' = 5 2 4 2 X; 0 -1 -2 -4 -2 x; (b) x' = (d) x' 1 X, 9 -6 6 x(0) = []; 4 -1 ; Sol'n 0 x, x(0) 4 3 -4. =
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
Please show all steps clearly, thank you. I only need help with problem 1.d.
![Exercises
1. Use Theorem 1 to obtain the general solution of the given first-order systems; if
initial conditions are given, find the particular solution satisfying them.
[23]
(a) x'
(c) x'
=
4
0
X;
1
-2
(b) x² = 6
-2 x; (d) x'
-4
1
4 x, x(0):
=
4 0
9
-6 -1
6
4
8₁
; Sol'n
0 x, x(0)
3
=
1
H](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13b8e596-45c0-434d-ab2b-ea5ebc953e71%2F1d25fff3-0f6a-436e-88e9-d28222969c65%2F0q7popw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercises
1. Use Theorem 1 to obtain the general solution of the given first-order systems; if
initial conditions are given, find the particular solution satisfying them.
[23]
(a) x'
(c) x'
=
4
0
X;
1
-2
(b) x² = 6
-2 x; (d) x'
-4
1
4 x, x(0):
=
4 0
9
-6 -1
6
4
8₁
; Sol'n
0 x, x(0)
3
=
1
H
Transcribed Image Text:Theorem 1. If A is an (n×n)-matrix of real constants that has an eigenbasis V₁, . . . , Vn
for R", then the general solution of (7.17) is given by
x(t) = C₁ e¹¹tv₁ + ... + C₂ e Ant√n₂
where A₁,..., An are the (not necessarily distinct) eigenvalues associated respectively
with the eigenvectors V₁,..., Vn.
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