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Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k >...
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)...
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...
dy 3. (5 points): Consider the autonomous differential equation dt is given below. Draw the phase...
dy 3. (5 points): Consider the autonomous differential equation dt is given below. Draw the phase line and classify the equilibria. f(y) where the graph of f(y) Y 1 -0.5 0.5 1 y
f 5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and...
f
5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and Gompertz equation: dy dt dy = (r-ay)y and where r,a>0 are constant dt a) For both, sketch the graph of f(y) versus y, find the critical points, and determine asymptotic stability. (b) For 0 y a, determine where the graph of y versus t is concave up and where it is concave down. (c) Sketch solution curves near critical points, discussing differences in the...
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are i...
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following autonomous first-order differential equation. dy =...
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following autonomous first-order differential equation. dy = y219 - y2) Find the critical points and phase portrait of the given differential equation. dx 6 3 3 0 0 ol -6 -6 -3 Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability, Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.) asymptotically stable...
2. Consider the nonlinear autonomous system of DEs: dx dt dy dt (a) Find all critical...
2. Consider the nonlinear autonomous system of DEs: dx dt dy dt (a) Find all critical points of this system. (Make sure that you have found all of them.) (b) Find the linearization (a linear system) at each critical point. Calculate the eigen- values of the contant coefficient matrix, classify the corresponding critical point, and state its stability.
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...
Consider the ODE dy/dt = -y(1-(y/T))(1-(y/K)) where 0 < T < K. (a) Draw a phase line and determine the stability o...
Consider the ODE dy/dt = -y(1-(y/T))(1-(y/K)) where 0 < T < K. (a) Draw a phase line and determine the stability of each equilibrium point. (b) Draw a graph in the yt-plane showing several solutions and their long term behavior. (Remark: Solving this ODE isn’t necessary!)
If a quantity y satisfies the differential equation dy = kx(10-y), k>0 dx. when X =...
If a quantity y satisfies the differential equation dy = kx(10-y), k>0 dx. when X = 2 and y = -7, the graph of yir increasing decreasing constant cannot be determined
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)...
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...
dy 3. (5 points): Consider the autonomous differential equation dt is given below. Draw the phase line and classify the equilibria. f(y) where the graph of f(y) Y 1 -0.5 0.5 1 y
f
5. (See 2.5, 16) Two models used in population modeling are the Logistic equation and Gompertz equation: dy dt dy = (r-ay)y and where r,a>0 are constant dt a) For both, sketch the graph of f(y) versus y, find the critical points, and determine asymptotic stability. (b) For 0 y a, determine where the graph of y versus t is concave up and where it is concave down. (c) Sketch solution curves near critical points, discussing differences in the...
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane
Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following autonomous first-order differential equation. dy = y219 - y2) Find the critical points and phase portrait of the given differential equation. dx 6 3 3 0 0 ol -6 -6 -3 Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability, Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.) asymptotically stable...
2. Consider the nonlinear autonomous system of DEs: dx dt dy dt (a) Find all critical points of this system. (Make sure that you have found all of them.) (b) Find the linearization (a linear system) at each critical point. Calculate the eigen- values of the contant coefficient matrix, classify the corresponding critical point, and state its stability.
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...
If a quantity y satisfies the differential equation dy = kx(10-y), k>0 dx. when X = 2 and y = -7, the graph of yir increasing decreasing constant cannot be determined
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