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Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines....
Differential Equations for Engineers II Page 1 of 6 1. The interface y(x) between air and...
Differential Equations for Engineers II Page 1 of 6 1. The interface y(x) between air and water in a time-independent open channel flow can be approximated with the second order ODE day d2 +oʻy=0, 20, (1) 1 mark 2 marks 5 marks where the parameter a? is a measure of the mean speed of the flow. The flow is in the positive x direction (i.e. from left to right). (a) Re-write equation (1) as a system of first-order ODEs by...
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 the system of equations dxdt=x(3−x−4y) dydt=y(1−3x), taking (x,y)>0. (1 point) Consider the system of equations...
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
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...
Consider the system x'=xy+y2 and y'=x2 -3y-4. Find all four equilibrium points and linearize the system...
Consider the system x'=xy+y2 and y'=x2 -3y-4. Find all four equilibrium points and linearize the system around each equilibrium point identifying it as a source, sink, saddle, spiral source or sink, center, or other. Find and sketch all nullclines and sketch the phase portrait. Show that the solution (x(t),y(t)) with initial conditions (x(0),y(0))=(-2.1,0) converges to an equilibrium point below the x axis and sketch the graphs of x(t) and y(t) on separate axes. Please write the answer on white paper...
dy -X dx2 dt =2y-x dt 2. Consider the following system of equations: phase plane, showing...
dy -X dx2 dt =2y-x dt 2. Consider the following system of equations: phase plane, showing only the first quadrant. (a) Graph the nullclines on a (b) Find the fixed points (there are two) to determine the nature of each fixed point (i.e., source, sink, saddle, and (c) Use Jacobian analysis whether it is a node or spiral). (d) Draw the flow arrows in each region of your phase plane from part (a). You may use a computer to help...
Please a- c for non linear system b 3. For each of the given non-linear systems,...
Please a- c for non linear system b
3. For each of the given non-linear systems, (a) find the equilibrium points, (b) near each equilibrium point, sketch the phase portrait of the linearized system, (c) use the information in (a) and (b) to sketch the phase portrait of the system: x' = - 4x + 4xy Sx = 2x – 2x² + 5xy ly=2y-y² – ry ly' = y - 2y2 + 2xy
Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and lin...
Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and linearize the system about each equilibrium point to find the type of the equilibrium point. b. Show that the system is a gradient system, and conclude that it has no periodic solutions. c. Sketch the phase portrait. Explain how you determined what the phase portrait looks like.
Consider the 2-dimensional system of linear equations -2 X' = 2 Note that the coefficient matrix...
Consider the 2-dimensional system of linear equations -2 X' = 2 Note that the coefficient matrix for this system contains a parameter a. (a) Determine the eigenvalues of the system in terms of a (b) The qualitative behavior of the solutions depends value ao where the qualitative behavior changes. Classify the equilibrium point of the system (by type and stability) when a < ao, when a = a), and when a > ao. on the value of a. Determine a...
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)...
Differential Equations for Engineers II Page 1 of 6 1. The interface y(x) between air and water in a time-independent open channel flow can be approximated with the second order ODE day d2 +oʻy=0, 20, (1) 1 mark 2 marks 5 marks where the parameter a? is a measure of the mean speed of the flow. The flow is in the positive x direction (i.e. from left to right). (a) Re-write equation (1) as a system of first-order ODEs by...
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 the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
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...
dy -X dx2 dt =2y-x dt 2. Consider the following system of equations: phase plane, showing only the first quadrant. (a) Graph the nullclines on a (b) Find the fixed points (there are two) to determine the nature of each fixed point (i.e., source, sink, saddle, and (c) Use Jacobian analysis whether it is a node or spiral). (d) Draw the flow arrows in each region of your phase plane from part (a). You may use a computer to help...
Please a- c for non linear system b
3. For each of the given non-linear systems, (a) find the equilibrium points, (b) near each equilibrium point, sketch the phase portrait of the linearized system, (c) use the information in (a) and (b) to sketch the phase portrait of the system: x' = - 4x + 4xy Sx = 2x – 2x² + 5xy ly=2y-y² – ry ly' = y - 2y2 + 2xy
Consider the 2-dimensional system of linear equations -2 X' = 2 Note that the coefficient matrix for this system contains a parameter a. (a) Determine the eigenvalues of the system in terms of a (b) The qualitative behavior of the solutions depends value ao where the qualitative behavior changes. Classify the equilibrium point of the system (by type and stability) when a < ao, when a = a), and when a > ao. on the value of a. Determine a...
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)...
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