[x =3x+2y y = 6x-y
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Graph the phase plane (xy-plane) for the following system of ODE’s using eigenvalues and eigenvectors.
![The image shows a system of differential equations, which can be a topic in the study of dynamical systems or differential equations. The system is written as follows:
\[
\begin{cases}
\dot{x} = 3x + 2y \\
\dot{y} = 6x - y
\end{cases}
\]
In this context, \(\dot{x}\) and \(\dot{y}\) represent the derivatives of \(x\) and \(y\) with respect to time, respectively. The equations describe how the variables \(x\) and \(y\) change over time, based on their current values.
This system could be used to analyze the behavior of two interacting quantities, such as populations in a predator-prey model or components in a mechanical system. Solving the system typically involves finding the equilibrium points and analyzing their stability, which provides insights into the long-term behavior of the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe82fa668-1fcd-4b5f-ba57-d40fab71d1eb%2F77a7267a-8ef2-4fa4-accf-e4888660f9c8%2F3zhw3g8_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a system of differential equations, which can be a topic in the study of dynamical systems or differential equations. The system is written as follows:
\[
\begin{cases}
\dot{x} = 3x + 2y \\
\dot{y} = 6x - y
\end{cases}
\]
In this context, \(\dot{x}\) and \(\dot{y}\) represent the derivatives of \(x\) and \(y\) with respect to time, respectively. The equations describe how the variables \(x\) and \(y\) change over time, based on their current values.
This system could be used to analyze the behavior of two interacting quantities, such as populations in a predator-prey model or components in a mechanical system. Solving the system typically involves finding the equilibrium points and analyzing their stability, which provides insights into the long-term behavior of the system.
Expert Solution
Step 1: Definition
Let X'=AX be a system of linear equations, then If the eigenvalues of A
- are positive , then critical point is unstable.
- are negative , then critical point is stable.
- have opposite sign , then critical point is saddle .
Step by step
Solved in 4 steps with 3 images
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