vector x' = [ the first row is 2 and 8, the second row is -1...
vector x' = [ the first row is 2 and 8, the second row is -1 and -2] vector x
(i) Compute the eigenvalues and eigenvectors of the system.
(ii) Use the eigenvalues to classify the equilibrium type of the origin.
(iii) Use the eigenvectors as guides to plot a phase portrait of the system.
(iv) Present a general solution to the system of ODE.
(v) Find the particular solution to this system of ODE if vector x(0) = [ the first row is 2, the second row is −1]. Identify which curve in the phase portrait corresponds to this solution.
(vi) If we dene vector x(t) = [ the first row is x(t), the second row is y(t) ], plot on their own set of axes the solutions x(t) and y(t) corresponding to your particular solution in the previous part.
Homework Answers
The first part of the question is just setting up the ODE as a couple equation in matrix form.
The first part of the question is just setting up the ODE as a
couple equation in matrix form.
Q1) Consider the ODE where y'(t), y"(t) denote respectively. an (c) Find the eigenvalues and eigenvectors of A and use these to plot the phase portrait for the system (2). (2 marks) (d) Does the system (2) obey the superposition principle? Explain. (2 marks)
Q1) Consider the ODE where y'(t), y"(t) denote respectively. an (c) Find the eigenvalues and eigenvectors of...
2. Consider the linear system: - (1 2) Y.with initial conditions) Y dt a) Compute the...
2. Consider the linear system: - (1 2) Y.with initial conditions) Y dt a) Compute the eigenvalues and eigenvectors for the system. b) For each eigenvalue, pick an associated eigenvector V and determine a vector solution y(t) to the system. c) Draw an accurate phase portrait for this system. What type of equilibrium point is the origin?
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1...
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1 1 3 (a) Determine the eigenvalues and eigenvectors of A. (b) Express the general solution of t' = Az in terms of real valued functions. (c) Sketch the phase portrait of the system. Do not forget to label your axes.
5.4 Equilibrium Solutions and Phase Portraits 1. 2 3 3 2 . (a) Draw direction field....
5.4 Equilibrium Solutions and Phase Portraits 1. 2 3 3 2 . (a) Draw direction field. Use the points: (0,0), (+1,0), (0, +1), (+1, +1). (b) Draw the phase portrait. (c) Classify the equilibrium solution with its stability. 11 and 2. Suppose 2 x 2 matrix A has eigenvalues – 3 and -1 with eigenvectors respectively. (a) Find the general solution of 7' = A. (b) Draw the phase portrait. (C) Classify the equilibrium solution with its stability. 3. Suppose...
Consider the linear system of first order differential equations x' = Ax, where x = x(t),...
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
1) Answer the following questions for harmonic oscillator with the given parameters and initial c...
1) Answer the following questions for harmonic oscillator with the given parameters and initial conditions Find the specific solution without converting to a linear system Convert to a linear system Find the eigenvalues and eigenvectors of the corresponding linear system Classify the oscillator (underdamped, overdamped, critically damped, undamped) (use technology to) Sketch the direction field and phase portrait Sketch the x(t)- and v(t)-graphs of the solution a. b. c. d. e. f. A) mass m-2, spring constant k 1, damping...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
I've figured out the first part but am struggling with the second
I've figured out the first part but am struggling with the
second
(1 point) Consider the linear system -5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. and λ2 = b. Find the real-valued solution to the initial value problem y( 334 + 2h, m(0) = 11, 2(0)15. = Use t as the independent variable in your answers. i(t) /2(t) 2 t)
(1 point) Consider the linear system -5 3 a. Find the eigenvalues and eigenvectors for...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
Problem 2. For the following system, (a) compute the eigenvalues, (b) compute the associated eigenvectors, (c)...
Problem 2. For the following system, (a) compute the eigenvalues, (b) compute the associated eigenvectors, (c) if the eigenvalues are complex, determine if the origin is a spiral sink, a spiral source, or a center; determine the natural period and natural frequency of the oscillations, and determine the direction of the oscillations in the phase plane, (d) sketch the phase portrait for the system; and (e) compute the general solution dY (1 -2
The first part of the question is just setting up the ODE as a
couple equation in matrix form.
Q1) Consider the ODE where y'(t), y"(t) denote respectively. an (c) Find the eigenvalues and eigenvectors of A and use these to plot the phase portrait for the system (2). (2 marks) (d) Does the system (2) obey the superposition principle? Explain. (2 marks)
Q1) Consider the ODE where y'(t), y"(t) denote respectively. an (c) Find the eigenvalues and eigenvectors of...
2. Consider the linear system: - (1 2) Y.with initial conditions) Y dt a) Compute the eigenvalues and eigenvectors for the system. b) For each eigenvalue, pick an associated eigenvector V and determine a vector solution y(t) to the system. c) Draw an accurate phase portrait for this system. What type of equilibrium point is the origin?
Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1 1 3 (a) Determine the eigenvalues and eigenvectors of A. (b) Express the general solution of t' = Az in terms of real valued functions. (c) Sketch the phase portrait of the system. Do not forget to label your axes.
5.4 Equilibrium Solutions and Phase Portraits 1. 2 3 3 2 . (a) Draw direction field. Use the points: (0,0), (+1,0), (0, +1), (+1, +1). (b) Draw the phase portrait. (c) Classify the equilibrium solution with its stability. 11 and 2. Suppose 2 x 2 matrix A has eigenvalues – 3 and -1 with eigenvectors respectively. (a) Find the general solution of 7' = A. (b) Draw the phase portrait. (C) Classify the equilibrium solution with its stability. 3. Suppose...
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
1) Answer the following questions for harmonic oscillator with the given parameters and initial conditions Find the specific solution without converting to a linear system Convert to a linear system Find the eigenvalues and eigenvectors of the corresponding linear system Classify the oscillator (underdamped, overdamped, critically damped, undamped) (use technology to) Sketch the direction field and phase portrait Sketch the x(t)- and v(t)-graphs of the solution a. b. c. d. e. f. A) mass m-2, spring constant k 1, damping...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
I've figured out the first part but am struggling with the
second
(1 point) Consider the linear system -5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. and λ2 = b. Find the real-valued solution to the initial value problem y( 334 + 2h, m(0) = 11, 2(0)15. = Use t as the independent variable in your answers. i(t) /2(t) 2 t)
(1 point) Consider the linear system -5 3 a. Find the eigenvalues and eigenvectors for...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
Problem 2. For the following system, (a) compute the eigenvalues, (b) compute the associated eigenvectors, (c) if the eigenvalues are complex, determine if the origin is a spiral sink, a spiral source, or a center; determine the natural period and natural frequency of the oscillations, and determine the direction of the oscillations in the phase plane, (d) sketch the phase portrait for the system; and (e) compute the general solution dY (1 -2
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