dx x(4 — х — 5у) dt dy = y(1 – 4x), dt taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: y=(4-x)/5 (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: y=4/(x+5) (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 = (0,0),(4,0),(1/4,3/4) (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 (5, 3), trajectories converge to v the point (-80,-57) (Enter the point as an (x,y) pair, e.g., (1,2).)
dx x(4 — х — 5у) dt dy = y(1 – 4x), dt taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: y=(4-x)/5 (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: y=4/(x+5) (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 = (0,0),(4,0),(1/4,3/4) (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 (5, 3), trajectories converge to v the point (-80,-57) (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
![### System of Differential Equations
We consider the system of differential equations:
\[
\frac{dx}{dt} = x(4 - x - 5y)
\]
\[
\frac{dy}{dt} = y(1 - 4x),
\]
taking \((x, y) > 0\).
#### (a) Nullclines
- **Vertical (x-) nullcline**: Write an equation for the non-zero vertical nullcline of this system:
\[
y = \frac{4-x}{5}
\]
(Enter your equation, e.g., \( y = x \).)
- **Horizontal (y-) nullcline**: Write an equation for the non-zero horizontal nullcline:
\[
y = \frac{4}{x+5}
\]
(Enter your equation, e.g., \( y = x \).)
(Note that there are also nullclines lying along the axes.)
#### (b) Equilibrium Points
What are the equilibrium points for the system?
Equilibria: \((0,0), (4,0), (1/4,3/4)\)
(Enter the points as comma-separated \((x, y)\) pairs, e.g., \((1,2), (3,4)\).)
#### (c) Estimating Trajectories
Use your nullclines to estimate trajectories in the phase plane, completing the following sentence:
If we start at the initial position \((5, 3)\), trajectories **converge to** the point \((-80,-57)\).
(Enter the point as an \((x, y)\) pair, e.g., \((1,2)\).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eee5fef-24c0-42d2-a99f-ec7ca4df2e97%2F7dd8d1b0-6110-4c5c-a548-58a3f3b71460%2Fxgh56ip_processed.png&w=3840&q=75)
Transcribed Image Text:### System of Differential Equations
We consider the system of differential equations:
\[
\frac{dx}{dt} = x(4 - x - 5y)
\]
\[
\frac{dy}{dt} = y(1 - 4x),
\]
taking \((x, y) > 0\).
#### (a) Nullclines
- **Vertical (x-) nullcline**: Write an equation for the non-zero vertical nullcline of this system:
\[
y = \frac{4-x}{5}
\]
(Enter your equation, e.g., \( y = x \).)
- **Horizontal (y-) nullcline**: Write an equation for the non-zero horizontal nullcline:
\[
y = \frac{4}{x+5}
\]
(Enter your equation, e.g., \( y = x \).)
(Note that there are also nullclines lying along the axes.)
#### (b) Equilibrium Points
What are the equilibrium points for the system?
Equilibria: \((0,0), (4,0), (1/4,3/4)\)
(Enter the points as comma-separated \((x, y)\) pairs, e.g., \((1,2), (3,4)\).)
#### (c) Estimating Trajectories
Use your nullclines to estimate trajectories in the phase plane, completing the following sentence:
If we start at the initial position \((5, 3)\), trajectories **converge to** the point \((-80,-57)\).
(Enter the point as an \((x, y)\) pair, e.g., \((1,2)\).)
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