Consider the differential equation dt = 2y(3-y)
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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Find the equilibria solutions
Plot the phase line
For each equilibrium solution, determine if it is asymptotically stable, semistable or unstable.
![Consider the differential equation
\[ \frac{dy}{dt} = 2y(3 - y) \]
Explanation of the problem: This equation is a first-order, non-linear differential equation that describes how the variable \( y \) changes with respect to the variable \( t \). The right-hand side of the equation \( 2y(3 - y) \) indicates that the rate of change of \( y \) depends on both \( y \) and the difference between 3 and \( y \).
If there are graphs or diagrams in the original content that illustrate the solutions or behavior of this differential equation, they typically show:
1. **Phase Line Diagram:**
- A graph with \( y \) on the horizontal axis and \( \frac{dy}{dt} \) on the vertical axis.
- The roots (where \( \frac{dy}{dt} = 0 \)) are \( y = 0 \) and \( y = 3 \). These are equilibrium points.
- Typically, arrows on the phase line indicate stability: movement towards or away from equilibrium points.
2. **Solution Curve:**
- A plot of \( y \) vs. \( t \) that illustrates how \( y \) evolves over time, given initial conditions.
- It can show the growth or the decline pattern depending on the initial value of \( y \).
3. **Direction Field:**
- A field of small arrows plotted in the \( y \)-\( t \) plane that shows the direction of \( \frac{dy}{dt} \) at each point.
- It helps visualize the trajectory of solutions for different initial conditions.
Understanding and solving this type of differential equation involves analyzing equilibrium points, stability, and using methods like separation of variables or integrating factors if applicable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47a605b-1167-4b41-bc40-883b330bf596%2F8dca54b8-cf2e-4f95-84da-563df538d60e%2Fn3jdot7_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the differential equation
\[ \frac{dy}{dt} = 2y(3 - y) \]
Explanation of the problem: This equation is a first-order, non-linear differential equation that describes how the variable \( y \) changes with respect to the variable \( t \). The right-hand side of the equation \( 2y(3 - y) \) indicates that the rate of change of \( y \) depends on both \( y \) and the difference between 3 and \( y \).
If there are graphs or diagrams in the original content that illustrate the solutions or behavior of this differential equation, they typically show:
1. **Phase Line Diagram:**
- A graph with \( y \) on the horizontal axis and \( \frac{dy}{dt} \) on the vertical axis.
- The roots (where \( \frac{dy}{dt} = 0 \)) are \( y = 0 \) and \( y = 3 \). These are equilibrium points.
- Typically, arrows on the phase line indicate stability: movement towards or away from equilibrium points.
2. **Solution Curve:**
- A plot of \( y \) vs. \( t \) that illustrates how \( y \) evolves over time, given initial conditions.
- It can show the growth or the decline pattern depending on the initial value of \( y \).
3. **Direction Field:**
- A field of small arrows plotted in the \( y \)-\( t \) plane that shows the direction of \( \frac{dy}{dt} \) at each point.
- It helps visualize the trajectory of solutions for different initial conditions.
Understanding and solving this type of differential equation involves analyzing equilibrium points, stability, and using methods like separation of variables or integrating factors if applicable.
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