13. r' = x² + x + 1
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Was needing some help on #13 and the proper way to create the phase line diagram. Does x=-1 and x=-2 seem correct?
. Draw the phase line for each equation and classify each equilibrium solution as a sink, a source, or a node.
### Differential Equations:
8. \( y' = 2y - 5 \)
9. \( \frac{dx}{dt} = (x-1)(x+2) \)
10. \( \frac{dx}{dt} = (x^2 - 1)(x - 2) \)
11. \( \frac{dy}{dx} = \sin 2y \)
12. \( x' = (x^2 + 1)(x - 1) \)
13. \( x' = x^2 + x + 1 \)
### Graphs and Diagrams:
For each of these equations, a phase line should be drawn to visually represent the behavior of the variables over time. Each phase line will indicate the equilibrium points and their nature (whether they are sinks, sources, or nodes).
- **Equilibrium Points:** These are the points where the derivative (or the right-hand side of the equation) equals zero. At these points, the system does not change.
- **Phase Line:** A visual tool used to illustrate the stability of equilibrium points. It shows the direction of the system's behavior (increasing or decreasing) around these points.
For example, if an equation has equilibrium solutions at \( x = a \) and \( x = b \):
- Draw a vertical line.
- Mark the equilibrium points \( x = a \) and \( x = b \).
- Indicate the direction of increase or decrease on either side of these points using arrows.
- Determine the stability:
- **Sink:** The solution moves towards the equilibrium (arrows point towards the equilibrium).
- **Source:** The solution moves away from the equilibrium (arrows point away from the equilibrium).
- **Node:** The nature can vary, and it may require further analysis.
**Note:** This is a summary for educational purposes, and students are encouraged to solve and plot each differential equation to fully understand the nature of the equilibrium points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe39b72c6-6c1d-4e20-8eee-b58971cb1bcd%2F634587dd-554c-454c-8ce1-6e77cf668fc1%2Fsouswrq_processed.png&w=3840&q=75)
Transcribed Image Text:## Equilibrium Solutions and Phase Lines
Find the equilibrium solutions for each of the differential equations in [Exercise Group 1.3.8.8—13](#). Draw the phase line for each equation and classify each equilibrium solution as a sink, a source, or a node.
### Differential Equations:
8. \( y' = 2y - 5 \)
9. \( \frac{dx}{dt} = (x-1)(x+2) \)
10. \( \frac{dx}{dt} = (x^2 - 1)(x - 2) \)
11. \( \frac{dy}{dx} = \sin 2y \)
12. \( x' = (x^2 + 1)(x - 1) \)
13. \( x' = x^2 + x + 1 \)
### Graphs and Diagrams:
For each of these equations, a phase line should be drawn to visually represent the behavior of the variables over time. Each phase line will indicate the equilibrium points and their nature (whether they are sinks, sources, or nodes).
- **Equilibrium Points:** These are the points where the derivative (or the right-hand side of the equation) equals zero. At these points, the system does not change.
- **Phase Line:** A visual tool used to illustrate the stability of equilibrium points. It shows the direction of the system's behavior (increasing or decreasing) around these points.
For example, if an equation has equilibrium solutions at \( x = a \) and \( x = b \):
- Draw a vertical line.
- Mark the equilibrium points \( x = a \) and \( x = b \).
- Indicate the direction of increase or decrease on either side of these points using arrows.
- Determine the stability:
- **Sink:** The solution moves towards the equilibrium (arrows point towards the equilibrium).
- **Source:** The solution moves away from the equilibrium (arrows point away from the equilibrium).
- **Node:** The nature can vary, and it may require further analysis.
**Note:** This is a summary for educational purposes, and students are encouraged to solve and plot each differential equation to fully understand the nature of the equilibrium points.
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