Consider the system of equations dx x(2 — х — Зу) dt dy = y(1 – 2x), dt taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (3, 1), trajectories ? v the point (Enter the point as an (x,y) pair, e.g., (1,2).)
Consider the system of equations dx x(2 — х — Зу) dt dy = y(1 – 2x), dt taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (3, 1), trajectories ? v the point (Enter the point as an (x,y) pair, e.g., (1,2).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the system of equations (see image) taking (x,y) > 0.
a. Write an equation for the (non-zero) vertical (x-)nullcline of this system. (Enter your equation, e.g., y=x.). And for the (non-zero) horizontal (y-)nullcline. (Enter your equation, e.g., y=x.)
b. What are the equilibrium points for the system? (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).)
c. Use your nullclines to estimate trajectories in the phase plane, completing the following sentence:
If we start at the initial position (3,1), trajectories [converge to, diverge from, cycle around, spiral into, spiral out from] the point (?, ?)
Transcribed Image Text:Consider the system of equations
dx
+ x(2 — х — Зу)
dt
dy
— У(1 — 2х),
dt
taking (x, y) > 0.
(a) Write an equation for the (non-zero) vertical (x-)nullcline of this system:
(Enter your equation, e.g., y=x.)
And for the (non-zero) horizontal (y-)nullcline:
(Enter your equation, e.g., y=x.)
(Note that there are also nullclines lying along the axes.)
(b) What are the equilibrium points for the system?
Equilibria =
(Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4),)
(c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence:
If we start at the initial position (3, 1), trajectories ?
v the point
(Enter the point as an (x,y) pair, e.g., (1,2).)
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