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.
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Consider the system: x' = y(1 + 2x) y' = x + x2 - y2 a. Find all the equilibrium points, and lin...
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...
Consider the following system: dx/dt=y(x^2+y^2-1) dy/dt= -x(x^2 +y^2-1) Find the equilibrium solution. 13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sk...
Consider the following system:
dx/dt=y(x^2+y^2-1)
dy/dt= -x(x^2 +y^2-1)
Find the equilibrium solution.
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
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 non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines....
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the system at each. Identify the type and sketch a local phase portrait (i.e. a sketch of the orbits just around the fixed points) at each fixed point. Show that the system has a time reversal symmetry. Draw a sketch showing isoclines and the directions of orbits in all parts of phase space. Use this information, together with the symmetry, to show there exists a...
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine...
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine the type of the equilibrium point at the origin. (35 points) (b) Find all straight-line solutions and draw the phase portrait for the system. (35 points) (c) What is the general solution to the system? (15 points) (d) Find the solution of the system with initial conditions: x(0) = 1 and y(0) = -1. (15 points)
Cal 4 , ) and use this to 6. Let f(x,y) = x2 + y2 + 2x + y. (a) Find all critical points of f in the disk {(x,y) : x2 + y2 < 4). Use the second derivative test to determine if these points corr...
Cal 4
, ) and use this to 6. Let f(x,y) = x2 + y2 + 2x + y. (a) Find all critical points of f in the disk {(x,y) : x2 + y2 < 4). Use the second derivative test to determine if these points correspond to a local maximum, local minimum, or saddle point. (b) Use Lagrange multipliers to find the absolute maximum/minimum values of f(x, y) on the circle a2 +y -4, as well as the points...
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)...
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 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 following system:
dx/dt=y(x^2+y^2-1)
dy/dt= -x(x^2 +y^2-1)
Find the equilibrium solution.
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
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 non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the system at each. Identify the type and sketch a local phase portrait (i.e. a sketch of the orbits just around the fixed points) at each fixed point. Show that the system has a time reversal symmetry. Draw a sketch showing isoclines and the directions of orbits in all parts of phase space. Use this information, together with the symmetry, to show there exists a...
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine the type of the equilibrium point at the origin. (35 points) (b) Find all straight-line solutions and draw the phase portrait for the system. (35 points) (c) What is the general solution to the system? (15 points) (d) Find the solution of the system with initial conditions: x(0) = 1 and y(0) = -1. (15 points)
Cal 4
, ) and use this to 6. Let f(x,y) = x2 + y2 + 2x + y. (a) Find all critical points of f in the disk {(x,y) : x2 + y2 < 4). Use the second derivative test to determine if these points correspond to a local maximum, local minimum, or saddle point. (b) Use Lagrange multipliers to find the absolute maximum/minimum values of f(x, y) on the circle a2 +y -4, as well as the points...
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)...
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...
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