Consider the non-linear autonomous system of differential equations dx dt dy dt =y-1 y x² + 2x - 1 (a) Find the two equilibria of the system. (b) For each equilibrium: i. Use linearisation to classify the equilibrium. ii. Sketch the phase portrait close to that equilibrium. You may find the following results useful: (2 1) (²1) = ²(2), (29) (-4)= -¹ (4)

Consider the non-linear autonomous system of differential equations dx dt dy dt =y-1 y x² + 2x - 1 (a) Find the two equilibria of the system. (b) For each equilibrium: i. Use linearisation to classify the equilibrium. ii. Sketch the phase portrait close to that equilibrium. You may find the following results useful: (2 1) (²1) = ²(2), (29) (-4)= -¹ (4)

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter11: Differential Equations

Section11.5: Nonlinear Systems Of Differential Equations

Problem 1YT: YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all...

See similar textbooks

Similar questions

YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all equilibrium points. b. Sketch a phase plane diagram, similar to Figure 11.
b) Given the differential equation system: (x' = -7x – 18y = 3x + 8y Find the general solution to the differential equation system. Sketch the phase plane of the differential equation system.
Let the system of differential equations be given by dx 2x(2x – 1) dt dy (x – 1)y dt (a) Find the x-isoclines and y-isoclines. (b) Write down the equilibrium points of the system. (c) Draw the phase diagram for the system. (d) Investigate the stability of the equilibrium points and interpret the phase diagram.
The survival of two species depends on their mutual cooperation. For instance, a species of bee that feeds primarily on the nectar of one plant species and simultaneously pollinates the plant. One simple model of mutualism is given by dx/xt = −ax + bxydy/dt = −my + nxy where a, b, m, and n are all positive constants. Perform a graphical analysis and indicate the trajectory directions in the phase plane.
Consider the autonomous system y' = y (y – 3) (a) Write a phase portrait for the above equation. (b) Identify any stable equilibrium solutions. (c) Identify any unstable equilibrium solutions.
characterize the equilibrium point for the system x′= Ax and sketch the phase portrait.
X: X(0) = -12 Draw the phase plane portrait, component graphs and state the equilibrium. (). Solve the system : X =/10 -5 2
Write the given third order linear equation as an equivalent system of first order equations with initial values. ƒ (1 − t²) y' — (4y"" + 3 cos(t)y") = −3 sin(t) - \y(3) = −3, _y'(3) = −2, y″(3) = -2 Use Y' Y1 = Y, with initial values Y Y2 = y', and (0) Y3 =y". Y+ 16 If you don't get this in 3 tries, you can get a hint.
1. (a) Consider the following equation: xy dy – (3x² + 4y²) dx = 0 (i) State the order of the equation, identify whether it's a linear or non-linear equation, and then state the type of equation. (ii) Solve the given equation.
A chemistry student heats a beaker that is at room temperature (30 ° C)by inserting it in an oven that has been preheated to 250 °C. If u is the temperature of the beaker in °C, what is the appropriate equation to model the temperature of the beaker. O du/dt = k(u –- 250) O du/dt = k(u – 30) O du/dt = k(250 – u) O du/dt = k(30 – u)
Find all equilibria for the following system of differential equations and determine the stability of each equilibrium. dx1 = 12x, (1- x1) - 6×,×2 dt dx2 =X2 (6- x2) - 5x2 dt ..... Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O A. The stable equilibria are (X1, X2 and the unstable equilibria are (x1, X2 (Type integers or fractions. Type ordered pairs. Use a comma to separate answers as needed.) O B. The stable equilibria are (x1, x2) =. There are no unstable equilibria. (Type an integer or a fraction. Type an ordered pair. Use a comma to separate answers as needed.) O C. The unstable equilibria are (x1. X2) = |. There are no stable equilibria. (Type an integer or a fraction. Type an ordered pair. Use a comma to separate answers as needed.) D. There are no equilibria.
Supposethat the system of differential equations describing a particular predator-prey model is given by dx/dt = 10-2 x - 2×10-5 xy; dy/dt = -4×10-2 y +10-5 xy where y and x are the numbers of predator and prey, respectively, and t is the time in days. (a) Determinethe numbers of predator and prey at equilibrium. (b) Write the equations of the phase curves in the neighborhood of the equilibrium point.