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Problem 2. For the following system, (a) compute the eigenvalues, (b) compute the associated eigenvectors, (c)...
Problem 3. 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 s...
Problem 3. 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. ar dY (1 -3 dt Y,
Problem 3. For the...
plesse show work 11. For the system dr 1 1 -3 -5 31 Y , initial...
plesse show work
11. For the system dr 1 1 -3 -5 31 Y , initial condition Y, - (4,0) Write the solution and sketch the x(i) and y(1) graphs of the particular solution If the eigenvalues are of the form a + ib, b0 then determine if the origin is a spiral sink, a spiral source, or a center determine the natural period and natural frequency of of the oscillations determine the directions of the oscillations in the phase...
4. (a) Write the corresponding first-order system, (b) find the eigenvalues and eigenvectors, (c)...
4. (a) Write the corresponding first-order system, (b) find the eigenvalues and eigenvectors, (c) classify the oscillator and, when appropriate, give the natural period, and (d) sketch the phase portrait. dt dt
4. (a) Write the corresponding first-order system, (b) find the eigenvalues and eigenvectors, (c) classify the oscillator and, when appropriate, give the natural period, and (d) sketch the phase portrait. dt dt
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that...
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that λ- (a) find the general solution; (b) determine if the origin is a spiral sink, a spiral source, or a center; (e) determine the direction of the oscillation in the phase plane (do the solutions go clockwise or countercdlocdkwise around the origin?); or counterclockwise
(3 points) Given the system 1. -2 0 2i and for the eigenvalue...
2 Y, Y(t) 2 dY 5. For the system dt - 2. a) Write the general...
2 Y, Y(t) 2 dY 5. For the system dt - 2. a) Write the general solution. b) State if the origin is a spiral sink, or a source, or a center. c) Write the natural period and the natural frequency of the solutions. d) Do the solutions go clockwise or anti clockwise around the origin? (0) e) Write the particular solution that corresponds to ly(0) =
5: For the following system, (a) provide the general solution, (b) provide a sketch showing the q...
5: For the following system, (a) provide the general solution, (b) provide a sketch showing the qualitative properties of the system, (c) specify what type ((spiral) source/sink, center, saddle) of equilibrium point the origin is. You should include any straight-line solutions as well as indicate more general types of solution curves. Be sure to indicate the direction of motion along any solution curve dY (1 -1
5: For the following system, (a) provide the general solution, (b) provide a sketch...
8. 20 pts.] Suppose that a 2 x2 matrix A has the following eigenvalues and eigenvectors:...
8. 20 pts.] Suppose that a 2 x2 matrix A has the following eigenvalues and eigenvectors: () 12, 1 r2=1, 2 2 (a) Classify the equilibrium 0 (node, saddle, spiral, center). Is it stable or unstable? (b) Sketch the trajectories of the system A , where a the phase plane below. (c) On the next page, sketch the graphs of r1 (t) and 2(t) versus t for the solution that satisfies the initial condition x1(0) = 1, x2(0) = 1...
a. Find the most general real-valued solution to the linear system of differential equations x =...
a. Find the most general real-valued solution to the linear system of differential equations x = -[42]; xid) + c2 x?(༧) b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these (1 point) Consider the linear system -6 7-11) -9 15 y. Find the eigenvalues and eigenvectors for the coefficient matrix. 21 = V1 = , and 12...
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors....
For each of the following systems:
(i) Find the general solution by using eigenvalues and
eigenvectors.
(ii) State whether the origin is stable, asymptotically stable,
or unstable.
(iii) State whether the origin is a node, saddle, center, or
spiral.
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors. (ii) State whether the origin is stable, asymptotically stable, or unstable. |(iii) State whether the origin is a node, saddle, center, or spiral. Problem 1:...
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?
Problem 3. 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. ar dY (1 -3 dt Y,
Problem 3. For the...
plesse show work
11. For the system dr 1 1 -3 -5 31 Y , initial condition Y, - (4,0) Write the solution and sketch the x(i) and y(1) graphs of the particular solution If the eigenvalues are of the form a + ib, b0 then determine if the origin is a spiral sink, a spiral source, or a center determine the natural period and natural frequency of of the oscillations determine the directions of the oscillations in the phase...
4. (a) Write the corresponding first-order system, (b) find the eigenvalues and eigenvectors, (c) classify the oscillator and, when appropriate, give the natural period, and (d) sketch the phase portrait. dt dt
4. (a) Write the corresponding first-order system, (b) find the eigenvalues and eigenvectors, (c) classify the oscillator and, when appropriate, give the natural period, and (d) sketch the phase portrait. dt dt
(3 points) Given the system 1. -2 0 2i and for the eigenvalue λ-2, the vector V-(1) is an eigenvector. we know that λ- (a) find the general solution; (b) determine if the origin is a spiral sink, a spiral source, or a center; (e) determine the direction of the oscillation in the phase plane (do the solutions go clockwise or countercdlocdkwise around the origin?); or counterclockwise
(3 points) Given the system 1. -2 0 2i and for the eigenvalue...
2 Y, Y(t) 2 dY 5. For the system dt - 2. a) Write the general solution. b) State if the origin is a spiral sink, or a source, or a center. c) Write the natural period and the natural frequency of the solutions. d) Do the solutions go clockwise or anti clockwise around the origin? (0) e) Write the particular solution that corresponds to ly(0) =
5: For the following system, (a) provide the general solution, (b) provide a sketch showing the qualitative properties of the system, (c) specify what type ((spiral) source/sink, center, saddle) of equilibrium point the origin is. You should include any straight-line solutions as well as indicate more general types of solution curves. Be sure to indicate the direction of motion along any solution curve dY (1 -1
5: For the following system, (a) provide the general solution, (b) provide a sketch...
8. 20 pts.] Suppose that a 2 x2 matrix A has the following eigenvalues and eigenvectors: () 12, 1 r2=1, 2 2 (a) Classify the equilibrium 0 (node, saddle, spiral, center). Is it stable or unstable? (b) Sketch the trajectories of the system A , where a the phase plane below. (c) On the next page, sketch the graphs of r1 (t) and 2(t) versus t for the solution that satisfies the initial condition x1(0) = 1, x2(0) = 1...
a. Find the most general real-valued solution to the linear system of differential equations x = -[42]; xid) + c2 x?(༧) b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these (1 point) Consider the linear system -6 7-11) -9 15 y. Find the eigenvalues and eigenvectors for the coefficient matrix. 21 = V1 = , and 12...
For each of the following systems:
(i) Find the general solution by using eigenvalues and
eigenvectors.
(ii) State whether the origin is stable, asymptotically stable,
or unstable.
(iii) State whether the origin is a node, saddle, center, or
spiral.
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors. (ii) State whether the origin is stable, asymptotically stable, or unstable. |(iii) State whether the origin is a node, saddle, center, or spiral. Problem 1:...
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?
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