a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A = ( ) -1 -1 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function ) = (CE)e-t is a solution of (1) if and only if 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 → t 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 line independent functions. h) use the chain rule to show that d* dy y x-5y = dx i) if y # 0 then O along the line x = 5y. How do you interpret this in the phase plane dy 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: (c1 + c2t)e and let (x(0), y(0) = (0,2). Show that x(t) = ty and t = - In () for> 0. C2
a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A = ( ) -1 -1 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function ) = (CE)e-t is a solution of (1) if and only if 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 → t 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 line independent functions. h) use the chain rule to show that d* dy y x-5y = dx i) if y # 0 then O along the line x = 5y. How do you interpret this in the phase plane dy 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: (c1 + c2t)e and let (x(0), y(0) = (0,2). Show that x(t) = ty and t = - In () for> 0. C2
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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