(ST-2) Find the critical points of the system below, and then use linearization to classify each one if possible. If it is not possible to classify a critical point this way, briefly explain why not. dx dt dy dt || || y(1 - x - y) x(3 − x - y)
(ST-2) Find the critical points of the system below, and then use linearization to classify each one if possible. If it is not possible to classify a critical point this way, briefly explain why not. dx dt dy dt || || y(1 - x - y) x(3 − x - y)
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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![**Problem (ST-2):**
Find the critical points of the system below, and then use linearization to classify each one if possible. If it is not possible to classify a critical point this way, briefly explain why not.
\[
\frac{dx}{dt} = y(1-x-y)
\]
\[
\frac{dy}{dt} = x(3-x-y)
\]
**Instructions:**
1. Identify the critical points of the given system of equations by setting \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\).
2. Use linearization to classify each critical point:
- Linearize the equations around each critical point.
- Determine the nature of each critical point using the Jacobian matrix and eigenvalues.
3. If linearization does not allow classification, explain why it is not possible.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3ec94bb-e469-4d7f-ad9d-5355d3a3cce0%2F99935497-67af-4ec4-82f1-226351e90b30%2Fukh7myc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem (ST-2):**
Find the critical points of the system below, and then use linearization to classify each one if possible. If it is not possible to classify a critical point this way, briefly explain why not.
\[
\frac{dx}{dt} = y(1-x-y)
\]
\[
\frac{dy}{dt} = x(3-x-y)
\]
**Instructions:**
1. Identify the critical points of the given system of equations by setting \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\).
2. Use linearization to classify each critical point:
- Linearize the equations around each critical point.
- Determine the nature of each critical point using the Jacobian matrix and eigenvalues.
3. If linearization does not allow classification, explain why it is not possible.
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