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Classify stability (i.e sync/source, spiral, saddle, center...) and sketch the phase portrait of the system ()-() 1 (а)...
1) Find the general solution of di = Ay where Then sketch the phase portrait in...
1) Find the general solution of di = Ay where Then sketch the phase portrait in the x-y plane, where Finally, classify the equilibrium solution at the origin as a source, spiral sink, etc. 2) Repeat for the matrix | 3 -31 -2 -2] 3) Repeat for the matrix 4 — 4) Repeat for the matrix [95 -9 15 but you don't need to sketch the phase portrait.
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...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the syst...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
The system of linear differential equations has the following solution e ( sitct)What is the type...
The system of linear differential equations has the following solution e ( sitct)What is the type of the phase portrait for sin 4t + 3 cos 4t this system? O spiral source nodal source spiral sink center O nodal sink saddle
1. (20 marks) This question is about the system of differential equations dY (3 1 (a) Consider the case k 0 i. Dete...
1. (20 marks) This question is about the system of differential equations dY (3 1 (a) Consider the case k 0 i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). i. Write down the general solution. ili Sketch a phase portrait for the system. (b) Now consider the case k -3. (-1+iv ) i. In this case, the matrix has an eigenvalue 2+i/2 with eigenvector and an eigenvalue 2-W2 with eigenvector Determine the type of equilibrium at...
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...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutio...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determina...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
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
Consider the system of differential equations ; 1) Find the fixed points of the system , 2) Eval...
consider the system of differential equations ; 1) Find the fixed points of the system , 2) Evaluate the Jacobian Matrix at each fixed point, 3) Classify stability of each fixed point, 4) Sketch the graph of the phase portrait,
1) Find the general solution of di = Ay where Then sketch the phase portrait in the x-y plane, where Finally, classify the equilibrium solution at the origin as a source, spiral sink, etc. 2) Repeat for the matrix | 3 -31 -2 -2] 3) Repeat for the matrix 4 — 4) Repeat for the matrix [95 -9 15 but you don't need to sketch the phase portrait.
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...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
The system of linear differential equations has the following solution e ( sitct)What is the type of the phase portrait for sin 4t + 3 cos 4t this system? O spiral source nodal source spiral sink center O nodal sink saddle
1. (20 marks) This question is about the system of differential equations dY (3 1 (a) Consider the case k 0 i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). i. Write down the general solution. ili Sketch a phase portrait for the system. (b) Now consider the case k -3. (-1+iv ) i. In this case, the matrix has an eigenvalue 2+i/2 with eigenvector and an eigenvalue 2-W2 with eigenvector Determine the type of equilibrium at...
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...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
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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