1. (20 points) Let (a) Determine and plot the equilibrium points and nullclines of the system....
1. (20 points) Let
(a) Determine and plot the equilibrium points and nullclines of
the system.
(b) Show the direction of the vector field between the
nullclines
(c) Sketch some solution curves starting near, but not on, the
equilibrium point(s).
(d) Label each equilibrium point as stable or unstable depending on
the behavior of the
solutions nearby, and describe the long-term behavior of all of the
solutions.
Homework Answers
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1. (20 points) Let
(a) Determine and plot the equilibrium points and nullclines of
the system....
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show...
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show the following: (a) Determine and plot the equilibrium points and nullclines. (b) Show the direction of the vector field between the nullclines as illustrated in Example 2 and Figure 2.6.3 in the textbook. (c) Sketch some solution curves starting near, but not on, the equilibrium points. (d) Label each equilibrium as stable or unstable depending on the behavior of the solutions that start nearby.
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...
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is...
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical...
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line.
4 Consider the autonomous differential equation y f(v)...
dP 7. For the equation = (P+2)(P2 - 6P+5)find the equilibrium points and make a phase...
dP 7. For the equation = (P+2)(P2 - 6P+5)find the equilibrium points and make a phase dt portrait of the differential equation. Classify each equilibrium point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves determined by the graphs of equilibrium solutions. (6pts)
For the equation (dp/dt)=(P+2)(P^2-6P+5) find the equilibrium points and make a phase portrait of the differential...
For the equation (dp/dt)=(P+2)(P^2-6P+5) find the equilibrium points and make a phase portrait of the differential equation. Classify each equilibrium point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves determined by the graphs of equilibrium solutions.
Polar bears and walruses have a complex dynamic relationship: the polar bears sometimes prey on the...
Polar bears and walruses have a complex dynamic relationship: the polar bears sometimes prey on the walruses, but multiple walruses together can fend off a polar bear attack, and in some cases even injure or kill the bear. The following system of differential equations attempts to model the populations of walruses (W) and polar bears (P): SW' = 14W – 2W2 – 3WP P' = 6P - 3p2 +8WP - 2W2P (a) Plot the nullclines of this system. Then use...
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. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7...
2. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7 points) Find all critical points and the phase portrait of the autonomous Ist order ODE dy dr -5y+4 Classify each critical point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves in the regions in the ry plane separated by equilibrium solutions. dy dx (S points) Solve the Bernoulli equation:-(- 31-1 7. (8 points) Solve the ODE by variation of parameters: -4y+4y (+...
1. (16) Consider the equation (a) (2) Determine all equilibrium solutions. (b) (6) Sketch a direction...
1. (16) Consider the equation (a) (2) Determine all equilibrium solutions. (b) (6) Sketch a direction field and describe the behavior of y as too if y(O) 1. (c) (8) Solve the equation exactly in explicit form.
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show the following: (a) Determine and plot the equilibrium points and nullclines. (b) Show the direction of the vector field between the nullclines as illustrated in Example 2 and Figure 2.6.3 in the textbook. (c) Sketch some solution curves starting near, but not on, the equilibrium points. (d) Label each equilibrium as stable or unstable depending on the behavior of the solutions that start nearby.
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...
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line.
4 Consider the autonomous differential equation y f(v)...
dP 7. For the equation = (P+2)(P2 - 6P+5)find the equilibrium points and make a phase dt portrait of the differential equation. Classify each equilibrium point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves determined by the graphs of equilibrium solutions. (6pts)
Polar bears and walruses have a complex dynamic relationship: the polar bears sometimes prey on the walruses, but multiple walruses together can fend off a polar bear attack, and in some cases even injure or kill the bear. The following system of differential equations attempts to model the populations of walruses (W) and polar bears (P): SW' = 14W – 2W2 – 3WP P' = 6P - 3p2 +8WP - 2W2P (a) Plot the nullclines of this system. Then use...
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. (8 points) Solve the linear, 1st order ODE with initial value: dy dr 3. (7 points) Find all critical points and the phase portrait of the autonomous Ist order ODE dy dr -5y+4 Classify each critical point as asymptotically stable, unstable or semi-stable. Sketch typical solution curves in the regions in the ry plane separated by equilibrium solutions. dy dx (S points) Solve the Bernoulli equation:-(- 31-1 7. (8 points) Solve the ODE by variation of parameters: -4y+4y (+...
1. (16) Consider the equation (a) (2) Determine all equilibrium solutions. (b) (6) Sketch a direction field and describe the behavior of y as too if y(O) 1. (c) (8) Solve the equation exactly in explicit form.
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