Consider the system of equations dæ = x(3 – x – 4y) dt dy = y(1 – 3æ), dt taking (x, y) > 0. dy = 0. Likewise, a vertical nullcline of this system is a line on which de = 0, and a horizontal Recall that a nullcline of this system is a line on which nullcline of this system is a line on which = 0. %3D (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 (4, 1), trajectories ? v the point (Enter the point as an (x.y) pair, e.g., (1,2).)
Consider the system of equations dæ = x(3 – x – 4y) dt dy = y(1 – 3æ), dt taking (x, y) > 0. dy = 0. Likewise, a vertical nullcline of this system is a line on which de = 0, and a horizontal Recall that a nullcline of this system is a line on which nullcline of this system is a line on which = 0. %3D (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 (4, 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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Question
Transcribed Image Text:Consider the system of equations
dæ
= x(3 – x – 4y)
dt
dy
= y(1 – 32),
dt
taking (x, y) > 0.
Recall that
nullcline of this system is a line on which de = 4 = 0. Likewise, a vertical nullcline of this system is a line on which de
= 0, and a horizontal
nullcline of this system is a line on which = 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 (4, 1), trajectories ?
v the point
(Enter the point as an (x,y) pair, e.g., (1,2).)
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