the following autonomous first order differential d_v=y² (36-y²) dx • Find critical points and phase portrait of the given differential ean. 1) consider equation. O shetch xy plane 12 -12 6 127 C ~ 12 clasify each critical points as asymptotically stable, unstable, or semi stable. (list the critical points according to their stability. asymptotically stable anstable Semi stable typical solution curves in the regions in
the following autonomous first order differential d_v=y² (36-y²) dx • Find critical points and phase portrait of the given differential ean. 1) consider equation. O shetch xy plane 12 -12 6 127 C ~ 12 clasify each critical points as asymptotically stable, unstable, or semi stable. (list the critical points according to their stability. asymptotically stable anstable Semi stable typical solution curves in the regions in
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
![**Problem Statement:**
1) Consider the following autonomous first order differential equation:
\[
\frac{dy}{dx} = y^2 (36 - y^2)
\]
- Find critical points and phase portrait of the given differential equation.
**Diagrams:**
- There are three separate diagrams illustrating the direction of solution curves (trajectories) around the critical points.
- The first diagram represents arrows going upwards between -6 and 0 and downwards on both sides of these points.
- The second diagram shows upward arrows between -6 and 6 and downward arrows on the outside.
- The third diagram presents upward arrows between 0 and 6 and downward on both external sides.
**Classification of Critical Points:**
Classify each critical point as asymptotically stable, unstable, or semi-stable. List the critical points according to their stability.
- Asymptotically Stable:
- Box provided for students to fill.
- Unstable:
- Box provided for students to fill.
- Semi-Stable:
- Box provided for students to fill.
**Additional Task:**
Sketch typical solution curves in the regions in the xy plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3827a2f7-6b05-43a2-b250-34175cac09a8%2Fa38ff676-fddb-42c8-93d2-db8145587877%2Ffzkr86_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1) Consider the following autonomous first order differential equation:
\[
\frac{dy}{dx} = y^2 (36 - y^2)
\]
- Find critical points and phase portrait of the given differential equation.
**Diagrams:**
- There are three separate diagrams illustrating the direction of solution curves (trajectories) around the critical points.
- The first diagram represents arrows going upwards between -6 and 0 and downwards on both sides of these points.
- The second diagram shows upward arrows between -6 and 6 and downward arrows on the outside.
- The third diagram presents upward arrows between 0 and 6 and downward on both external sides.
**Classification of Critical Points:**
Classify each critical point as asymptotically stable, unstable, or semi-stable. List the critical points according to their stability.
- Asymptotically Stable:
- Box provided for students to fill.
- Unstable:
- Box provided for students to fill.
- Semi-Stable:
- Box provided for students to fill.
**Additional Task:**
Sketch typical solution curves in the regions in the xy plane.
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