Write each system in matrix form- = Ax. dx dt Determine the eigenvalues of A. State whether the origin is a stable or unstable e State whether the origin is a node, saddle point, s State the equations of the straight-line trajectorie owards or away from the origin. If none exist, st A has real eigenvalues, then determine the eige olve the system. (See examples in Section 7.4) x = x + 4y It y It = 4x + y
Write each system in matrix form- = Ax. dx dt Determine the eigenvalues of A. State whether the origin is a stable or unstable e State whether the origin is a node, saddle point, s State the equations of the straight-line trajectorie owards or away from the origin. If none exist, st A has real eigenvalues, then determine the eige olve the system. (See examples in Section 7.4) x = x + 4y It y It = 4x + y
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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Parts D, E, and F. I added the photo that has the first parts completed.
Transcribed Image Text:For the following systems, the origin is the equilibrium point.
dx
a) Write each system in matrix form = Ax.
dt
b)
Determine the eigenvalues of A.
c) State whether the origin is a stable or unstable equilibrium.
d)
State whether the origin is a node, saddle point, spiral point, or center.
e) State the equations of the straight-line trajectories and tell whether they are going
towards or away from the origin. If none exist, state so.
f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to
solve the system. (See examples in Section 7.4)
4.
dx
dt
dy
dt
= x + 4y
= 4x + y
![dy
4a) de
= x+4y
= 4x+y
b) [1->
< unstable
4
[*2] = [41] [4] A= [41]
(1-7) ²
2²-2x-15=0
(7-5) (x+3) =0
equilibrium
[5 and
M](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24638e23-7712-442e-8ea2-f441f488d4c1%2F3eaf28ca-23ff-49e0-8e2d-c05e589b3ad8%2Fzadmel6_processed.png&w=3840&q=75)
Transcribed Image Text:dy
4a) de
= x+4y
= 4x+y
b) [1->
< unstable
4
[*2] = [41] [4] A= [41]
(1-7) ²
2²-2x-15=0
(7-5) (x+3) =0
equilibrium
[5 and
M
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