x' = Ax = x with x(0) = where x = [x1,x2]". (a) Find the eigenvalues of the matrix of coefficients A. They should be real and distinct. Call the larger one A and the smaller one A2, ie A1 > d2. (b) Find the eigenvectors associated with these eigenvalues. Call them v1) and v2) respectively. (c) The general solution is of the form x(t) = cje'v(1) + Write down the general solution for this case. (d) Is the equilibrium point for this system stable or unstable? (e) Is the equilibrium point here a node, a saddle or a spiral? (f) By applying the initial condition, find the coefficients c1 and c2. (g) Write out your answer for the particular solution for this initial value problem. (h) Show that your solution is correct, ie that is satisfies the initial value problem. x1(t) (i) For long times, evaluate the limit: lim 1400 X2(t)
x' = Ax = x with x(0) = where x = [x1,x2]". (a) Find the eigenvalues of the matrix of coefficients A. They should be real and distinct. Call the larger one A and the smaller one A2, ie A1 > d2. (b) Find the eigenvectors associated with these eigenvalues. Call them v1) and v2) respectively. (c) The general solution is of the form x(t) = cje'v(1) + Write down the general solution for this case. (d) Is the equilibrium point for this system stable or unstable? (e) Is the equilibrium point here a node, a saddle or a spiral? (f) By applying the initial condition, find the coefficients c1 and c2. (g) Write out your answer for the particular solution for this initial value problem. (h) Show that your solution is correct, ie that is satisfies the initial value problem. x1(t) (i) For long times, evaluate the limit: lim 1400 X2(t)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
Related questions
Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
Find d-f
![x' = Ax =
x with x(0)
where x =
[x1, x2]".
(a) Find the eigenvalues of the matrix of coefficients A. They should be real and distinct. Call the larger one d1 and the
smaller one A2, ie A1 > A2.
(b) Find the eigenvectors associated with these eigenvalues. Call them v(1) and v2) respectively.
(c) The general solution is of the form
x(t) = cjei'v(1) + c2e*z'y(2) .
Write down the general solution for this case.
(d) Is the equilibrium point for this system stable or unstable?
(e) Is the equilibrium point here a node, a saddle or a spiral?
(f) By applying the initial condition, find the coefficients c and c2.
(g) Write out your answer for the particular solution for this initial value problem.
(h) Show that your solution is correct, ie that is satisfies the initial value problem.
x1(t)
(i) For long times, evaluate the limit: lim
1-0 x2(t)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b5c4bb0-0c91-4b12-a9d0-169013e798bf%2F5d128621-748d-4dc9-809a-8a3fa8c9747b%2Fahet4cn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:x' = Ax =
x with x(0)
where x =
[x1, x2]".
(a) Find the eigenvalues of the matrix of coefficients A. They should be real and distinct. Call the larger one d1 and the
smaller one A2, ie A1 > A2.
(b) Find the eigenvectors associated with these eigenvalues. Call them v(1) and v2) respectively.
(c) The general solution is of the form
x(t) = cjei'v(1) + c2e*z'y(2) .
Write down the general solution for this case.
(d) Is the equilibrium point for this system stable or unstable?
(e) Is the equilibrium point here a node, a saddle or a spiral?
(f) By applying the initial condition, find the coefficients c and c2.
(g) Write out your answer for the particular solution for this initial value problem.
(h) Show that your solution is correct, ie that is satisfies the initial value problem.
x1(t)
(i) For long times, evaluate the limit: lim
1-0 x2(t)
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