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2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A- () 5

2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A- () 5

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A - ) 5 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function (*DE)e-t is a solution of (1) if and only if .c+dt A (") - C-) A and = - c) use the above exercises to find the general solution d) Asses whether the solution you got consist of two linearly independent functions. e) explain why the solution curves, when t → t0 are parallel with x-axis f) find the solution curve through (x(to), y(to)) = (xo,0), for a given x, + 0 g) find the tangent direction of the solution curves when intersecting the y-axis of two linearly independent functions. dx x-5y = h) use the chain rule to show that- dy у dx i) if y + 0 then dy O along the line x 5y. How do you interpret this in the phase plane? j) make a drawing of the solution curve in the phase plane k) what type of critical point is the origin and determine the stability of this. I) consider the curves given by: G)-(* (c1 + c2t)e 5y and let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In 1(2) for> 0. C2
Transcribed Image Text:2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A - ) 5 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function (*DE)e-t is a solution of (1) if and only if .c+dt A (") - C-) A and = - c) use the above exercises to find the general solution d) Asses whether the solution you got consist of two linearly independent functions. e) explain why the solution curves, when t → t0 are parallel with x-axis f) find the solution curve through (x(to), y(to)) = (xo,0), for a given x, + 0 g) find the tangent direction of the solution curves when intersecting the y-axis of two linearly independent functions. dx x-5y = h) use the chain rule to show that- dy у dx i) if y + 0 then dy O along the line x 5y. How do you interpret this in the phase plane? j) make a drawing of the solution curve in the phase plane k) what type of critical point is the origin and determine the stability of this. I) consider the curves given by: G)-(* (c1 + c2t)e 5y and let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In 1(2) for> 0. C2
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