2i) 2.-a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A.A - )5b) consider the system:(x' = -x + 5yy' = -y(1)Show that the following vector function(*DE)e-t is a solution of (1) if and only if.c+dtA (") - C-)Aand= -c) use the above exercises to find the general solutiond) 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-axisf) find the solution curve through (x(to), y(to)) = (xo,0), for a given x, + 0g) find the tangent direction of the solution curves when intersecting the y-axis of two linearlyindependent functions.dxx-5y=h) use the chain rule to show that-dyуdxi) if y + 0 thendyO along the line x5y. How do you interpret this in the phase plane?j) make a drawing of the solution curve in the phase planek) what type of critical point is the origin and determine the stability of this.I) consider the curves given by:G)-(*(c1 + c2t)e5yand let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In1(2) for> 0.C2

Since there are multiple questions the first question is answered as per our guidelines.Kindly…

2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A = -1 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function (*) = (*+bE)e-t is a solution of (1) if and only if c+dt A () - C=) A and \d 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 → to are parallel with x-axis f) find the solution curve through (x(t,), 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. x-5y dx h) use the chain rule to show that dy y dx i) if y # 0 then O along the line x = 5y. How do you interpret this in the phase plane? dy %3D 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¬t' e-t 5y and let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In( () for > 0. C2

2.- a) find eigenvalue/eigenvalues and the corresponding eigenvector/eigenvectors of A. A = -1 b) consider the system: (x' = -x + 5y y' = -y (1) Show that the following vector function () = (+bE)e-t is a solution of (1) if and only if c+dt A () - C=) A and \d 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 → to are parallel with x-axis f) find the solution curve through (x(t,), 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. x-5y dx h) use the chain rule to show that dy y dx i) if y # 0 then O along the line x = 5y. How do you interpret this in the phase plane? dy %3D 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¬t' e-t 5y and let (x(0), y(0)) = (0,2). Show that x(t) = ty and t = – In( () for > 0. C2

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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