Sketch a directional field for the equation. Identify at least 3 isoclines. dy dx =y-x+2
Sketch a directional field for the equation. Identify at least 3 isoclines. dy dx =y-x+2
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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Question
![**Educational Content on Directional Fields and Isoclines**
**Task Description:**
- Sketch a directional field for the equation. Identify at least 3 isoclines.
**Equation:**
\[
\frac{dy}{dx} = y - x + 2
\]
**Graphical Explanation:**
The image includes a set of axes prepared for graphing, with a vertical and a horizontal axis intersecting each other at right angles. This setup is intended for sketching the directional field of the differential equation provided.
**Key Concepts:**
- **Directional Field**: A graphical representation of the slopes of a differential equation at various points in the plane. Each segment in the field indicates the slope of the solution at that point.
- **Isoclines**: Lines in the slope field where the slope (given by \(\frac{dy}{dx}\)) is constant. For this equation, isoclines can be found by setting \(y - x + 2 = c\), where \(c\) is a constant.
**Instructions:**
1. Plot points on the graph where you calculate the slope \(\frac{dy}{dx} = y - x + 2\).
2. Draw short line segments at these points with slopes corresponding to the calculated values.
3. Determine at least three isoclines by choosing different values for \(c\) and solving \(y - x + 2 = c\) for lines like \(y = x + c - 2\).
**Example Isoclines:**
- If \(c = 0\), the isocline is \(y = x - 2\).
- If \(c = 1\), the isocline is \(y = x - 1\).
- If \(c = -1\), the isocline is \(y = x - 3\).
By plotting these isoclines and the directional field, you will have a clearer understanding of how solutions to the differential equation behave in the plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F63b17597-09c9-45cc-9667-49c7224e6bc2%2Fnsupboc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Directional Fields and Isoclines**
**Task Description:**
- Sketch a directional field for the equation. Identify at least 3 isoclines.
**Equation:**
\[
\frac{dy}{dx} = y - x + 2
\]
**Graphical Explanation:**
The image includes a set of axes prepared for graphing, with a vertical and a horizontal axis intersecting each other at right angles. This setup is intended for sketching the directional field of the differential equation provided.
**Key Concepts:**
- **Directional Field**: A graphical representation of the slopes of a differential equation at various points in the plane. Each segment in the field indicates the slope of the solution at that point.
- **Isoclines**: Lines in the slope field where the slope (given by \(\frac{dy}{dx}\)) is constant. For this equation, isoclines can be found by setting \(y - x + 2 = c\), where \(c\) is a constant.
**Instructions:**
1. Plot points on the graph where you calculate the slope \(\frac{dy}{dx} = y - x + 2\).
2. Draw short line segments at these points with slopes corresponding to the calculated values.
3. Determine at least three isoclines by choosing different values for \(c\) and solving \(y - x + 2 = c\) for lines like \(y = x + c - 2\).
**Example Isoclines:**
- If \(c = 0\), the isocline is \(y = x - 2\).
- If \(c = 1\), the isocline is \(y = x - 1\).
- If \(c = -1\), the isocline is \(y = x - 3\).
By plotting these isoclines and the directional field, you will have a clearer understanding of how solutions to the differential equation behave in the plane.
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