a. Consider the system of differential equations given by: -3 34 y, Y1(0) y = -1 where some initial conditions are given. You are given that this system generates a CENTER. There is a parameter, a, in the matrix, which you must find, so that this system is a CENTER. a = Find the eigenvalues and eigenvectors of the matrix above with the appropriate a. For the complex values use į to represent the imaginary part. and 2 %3D In the written work show the key steps required to find the eigenvalues and eigenvectors (not how a computer program finds them). Also, give the general real-valued solution to the linear system of differential equations. b. Find the unique solution to the initial value problem in y(t) given above (where t is the independent variable) and in your written work show how you find the arbitrary constants (without a computer): Y1(t) = Y2(t)
a. Consider the system of differential equations given by: -3 34 y, Y1(0) y = -1 where some initial conditions are given. You are given that this system generates a CENTER. There is a parameter, a, in the matrix, which you must find, so that this system is a CENTER. a = Find the eigenvalues and eigenvectors of the matrix above with the appropriate a. For the complex values use į to represent the imaginary part. and 2 %3D In the written work show the key steps required to find the eigenvalues and eigenvectors (not how a computer program finds them). Also, give the general real-valued solution to the linear system of differential equations. b. Find the unique solution to the initial value problem in y(t) given above (where t is the independent variable) and in your written work show how you find the arbitrary constants (without a computer): Y1(t) = Y2(t)
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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Transcribed Image Text:a. Consider the system of differential equations given by:
-3
34
y,
Y1(0)
-1
-a
where some initial conditions are given. You are given that this system generates a CENTER. There is a parameter, a, in the
matrix, which you must find, so that this system is a CENTER.
a =
Find the eigenvalues and eigenvectors of the matrix above with the appropriate a. For the complex values use į to represent the
imaginary part.
and 2
In the written work show the key steps required to find the eigenvalues and eigenvectors (not how a computer program finds
them). Also, give the general real-valued solution to the linear system of differential equations.
b. Find the unique solution to the initial value problem in y(t) given above (where t is the independent variable) and in your written
work show how you find the arbitrary constants (without a computer):
Y1(t) =
Y2(t)
c. In the written part of this problem include a reasonable Phase Portrait. The Phase Portrait needs to include the equilibrium, show
the position and direction of flow for all real eigenvectors or show the direction of flow with clockwise or counter-clockwise flow
for complex eigenvalue problems. Include several typical trajectories (at least 4). Recall that this is a CENTER.
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