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Problem 1. For each of the following systems of autonomous differential equations, sketch the nul...
3. Consider the following system of autonomous differential equations for the populations of two ...
Please solve the following problem, solve all parts
3. Consider the following system of autonomous differential equations for the populations of two species: dx dt dy dt --0.2y0.0004 ry 0.1 x 0.001 ry a) What type of system might this represent (and why) ? b) Are there equilibria? If yes, what are they? c) Perform a graphical analysis and sketch some trajectories in the phase plane. Comment on the stability of any equilibria. d) What would you predict for the...
Problem 3 (Stewart & Day 7.2.9) For each of autonomous differential equations below, find all equilibria...
Problem 3 (Stewart & Day 7.2.9) For each of autonomous differential equations below, find all equilibria and determine the values of a for which each equilibrium is locally stable. Assume that a #0 for all equations. (a) y = 1+ ay. (b) y = 1- e-ay. (c) y' = aecos(y).
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by...
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by examining dt the equilibrium solutions of the equation for various values of the parameter 8 1. Find the equilibrium solutions of the equation for 8 = -4,-2, 0, 2, 4 and make a sketch of the phase line for each value. Determine the stability of each equilibria. 2. Use a computer or some other means to sketch some solution curves for each value of...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature o...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. (b) x' = x(y + 2x-2), y' = y(y + x-3)
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt = 3xy − y 3a (5 pts): Find the steady states of the system. 3b (15 pts): Linearize the model about each of the fixed points and determine the type of stability. 3b (15 pts): Draw the phase portrait for this system, including nullclines, flow trajectories, and all fixed points. Problem 2 (25 pts): Two-dimensional linear ODEs For the following linear systems, identify the...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature o...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. x' = x(y + 2x-2), y' = y(y-1 ) (a)
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for...
for differential equations 1. Identify each of the following differential equations as either Separable, Homogeneous, Linear...
for
differential equations
1. Identify each of the following differential equations as either Separable, Homogeneous, Linear Bernoulli, or Exact and solve the equation using the method of the type you have identified. Many can be classified in multiple ways, it is not necessary to list all possibilities. (3xy2 +2ycos x)+y'-y sin x-x =0 Туре: A. dx General Solution: B. (4xy+xy)2x+ xy2 dx Туре: General Solution: Туре: C. y'y'y+1 General Solution: (3x'y+e')-(2y-x-xe)dy Туре: D. dx General Solution: Туре: dy E. =y(xy-1)...
nullcline example: 1. For each of the following systems, sketch the x- and y-nullclines and use this information to de...
nullcline example:
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. (b) x' = x(y + 2x-2), y' = y(y + x-3) Figure 9.4 The (a) nullclines and (b) phase portrait for x'
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2...
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively
15 pts] Sketch some representative...
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...
Please solve the following problem, solve all parts
3. Consider the following system of autonomous differential equations for the populations of two species: dx dt dy dt --0.2y0.0004 ry 0.1 x 0.001 ry a) What type of system might this represent (and why) ? b) Are there equilibria? If yes, what are they? c) Perform a graphical analysis and sketch some trajectories in the phase plane. Comment on the stability of any equilibria. d) What would you predict for the...
Problem 3 (Stewart & Day 7.2.9) For each of autonomous differential equations below, find all equilibria and determine the values of a for which each equilibrium is locally stable. Assume that a #0 for all equations. (a) y = 1+ ay. (b) y = 1- e-ay. (c) y' = aecos(y).
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by examining dt the equilibrium solutions of the equation for various values of the parameter 8 1. Find the equilibrium solutions of the equation for 8 = -4,-2, 0, 2, 4 and make a sketch of the phase line for each value. Determine the stability of each equilibria. 2. Use a computer or some other means to sketch some solution curves for each value of...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. (b) x' = x(y + 2x-2), y' = y(y + x-3)
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only...
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. x' = x(y + 2x-2), y' = y(y-1 ) (a)
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for...
for
differential equations
1. Identify each of the following differential equations as either Separable, Homogeneous, Linear Bernoulli, or Exact and solve the equation using the method of the type you have identified. Many can be classified in multiple ways, it is not necessary to list all possibilities. (3xy2 +2ycos x)+y'-y sin x-x =0 Туре: A. dx General Solution: B. (4xy+xy)2x+ xy2 dx Туре: General Solution: Туре: C. y'y'y+1 General Solution: (3x'y+e')-(2y-x-xe)dy Туре: D. dx General Solution: Туре: dy E. =y(xy-1)...
nullcline example:
1. For each of the following systems, sketch the x- and y-nullclines and use this information to determine the nature of the phase portrait. You may assume that these systems are defined only for x,y 20. (b) x' = x(y + 2x-2), y' = y(y + x-3) Figure 9.4 The (a) nullclines and (b) phase portrait for x'
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively
15 pts] Sketch some representative...
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...
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