Find and classify the fixed points of 0 = sin 0 + cos 0, and sketch the phase portrait both as vs. 0, and on the circle.
Find and classify the fixed points of 0 = sin 0 + cos 0, and sketch the phase portrait both as vs. 0, and on the circle.
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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![### Mathematical Analysis of Fixed Points
#### Problem Statement
Find and classify the fixed points of the differential equation:
\[ \dot{\theta} = \sin \theta + \cos \theta \]
Additionally, sketch the phase portrait in two ways:
1. As \( \dot{\theta} \) versus \( \theta \)
2. On the circle
#### Explanation
To determine the fixed points, set \( \dot{\theta} = 0 \), which gives the equation:
\[ \sin \theta + \cos \theta = 0 \]
By solving the above equation, you can determine the values of \( \theta \) that satisfy this condition, which will be the fixed points. Classify these fixed points as stable, unstable, or semi-stable based on the behavior of \( \dot{\theta} \) near these points.
#### Sketching Phase Portraits
1. **\( \dot{\theta} \) vs. \( \theta \) Plot:**
- Plot the function \( \dot{\theta} = \sin \theta + \cos \theta \) as a graph to visualize how \( \dot{\theta} \) changes with \( \theta \).
- The x-axis represents \( \theta \) and the y-axis represents \( \dot{\theta} \).
- Identify and label the fixed points and the stability by observing where the curve crosses the x-axis.
2. **Phase Portrait on the Circle:**
- Represent \( \theta \) on a circular axis, reflecting the periodic nature of the trigonometric functions.
- Mark the direction of trajectories on the circle indicating increasing or decreasing \( \theta \).
- Use arrows to show the flow around fixed points, clearly marking stable and unstable points.
This exercise facilitates understanding of dynamic systems, fixed point analysis, and phase portraits in mathematical studies of differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F101ed911-f56b-49a4-b759-448eacbeb418%2F2k7nb4_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Analysis of Fixed Points
#### Problem Statement
Find and classify the fixed points of the differential equation:
\[ \dot{\theta} = \sin \theta + \cos \theta \]
Additionally, sketch the phase portrait in two ways:
1. As \( \dot{\theta} \) versus \( \theta \)
2. On the circle
#### Explanation
To determine the fixed points, set \( \dot{\theta} = 0 \), which gives the equation:
\[ \sin \theta + \cos \theta = 0 \]
By solving the above equation, you can determine the values of \( \theta \) that satisfy this condition, which will be the fixed points. Classify these fixed points as stable, unstable, or semi-stable based on the behavior of \( \dot{\theta} \) near these points.
#### Sketching Phase Portraits
1. **\( \dot{\theta} \) vs. \( \theta \) Plot:**
- Plot the function \( \dot{\theta} = \sin \theta + \cos \theta \) as a graph to visualize how \( \dot{\theta} \) changes with \( \theta \).
- The x-axis represents \( \theta \) and the y-axis represents \( \dot{\theta} \).
- Identify and label the fixed points and the stability by observing where the curve crosses the x-axis.
2. **Phase Portrait on the Circle:**
- Represent \( \theta \) on a circular axis, reflecting the periodic nature of the trigonometric functions.
- Mark the direction of trajectories on the circle indicating increasing or decreasing \( \theta \).
- Use arrows to show the flow around fixed points, clearly marking stable and unstable points.
This exercise facilitates understanding of dynamic systems, fixed point analysis, and phase portraits in mathematical studies of differential equations.
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