Find the critical point set for the given system. dx = x-y dt dy 7x2 + 9y? - 5 %3D dt Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The set of critical points is (Use a comma to separate answers as needed. Type an ordered pair. Type an exact answer, using radicals as needed.) O B. There are no critical points.
Find the critical point set for the given system. dx = x-y dt dy 7x2 + 9y? - 5 %3D dt Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The set of critical points is (Use a comma to separate answers as needed. Type an ordered pair. Type an exact answer, using radicals as needed.) O B. There are no critical points.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Analyzing Critical Points for a Dynamical System**
**Introduction:**
Understanding the critical points of a dynamical system can reveal essential information about its behavior. This exercise involves finding these critical points for a given system of differential equations.
**Problem Statement:**
Determine the critical point set for the following system:
\[
\frac{dx}{dt} = x - y
\]
\[
\frac{dy}{dt} = 7x^2 + 9y^2 - 5
\]
**Instructions:**
Select the correct choice and, if necessary, fill in the answer box to complete your choice.
**Options:**
- **A.** The set of critical points is \( \{ \, \} \).
- (Use a comma to separate answers as needed. Type an ordered pair. Type an exact answer, using radicals as needed.)
- **B.** There are no critical points.
**Explanation:**
The critical points occur where both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). Calculating these points involves solving the equations simultaneously:
\[
x - y = 0
\]
\[
7x^2 + 9y^2 - 5 = 0
\]
This process often involves algebraic manipulation or the use of computational tools. Understanding and identifying these points help in analyzing the stability and behavior of the system.
**Conclusion:**
After evaluating the expressions, enter the ordered pairs for the critical points or confirm if there are none. This knowledge is foundational for deeper studies in differential equations and dynamical systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d735952-8e6c-4892-9968-89cd884eae37%2F2bc66605-7530-4553-a8e4-556ca3c6a9e4%2Firlr1kgk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Critical Points for a Dynamical System**
**Introduction:**
Understanding the critical points of a dynamical system can reveal essential information about its behavior. This exercise involves finding these critical points for a given system of differential equations.
**Problem Statement:**
Determine the critical point set for the following system:
\[
\frac{dx}{dt} = x - y
\]
\[
\frac{dy}{dt} = 7x^2 + 9y^2 - 5
\]
**Instructions:**
Select the correct choice and, if necessary, fill in the answer box to complete your choice.
**Options:**
- **A.** The set of critical points is \( \{ \, \} \).
- (Use a comma to separate answers as needed. Type an ordered pair. Type an exact answer, using radicals as needed.)
- **B.** There are no critical points.
**Explanation:**
The critical points occur where both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). Calculating these points involves solving the equations simultaneously:
\[
x - y = 0
\]
\[
7x^2 + 9y^2 - 5 = 0
\]
This process often involves algebraic manipulation or the use of computational tools. Understanding and identifying these points help in analyzing the stability and behavior of the system.
**Conclusion:**
After evaluating the expressions, enter the ordered pairs for the critical points or confirm if there are none. This knowledge is foundational for deeper studies in differential equations and dynamical systems.
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