Sketch a directional field for the equation. Identify at least 4 isoclines. dy dx = y +x Isoclines:
Sketch a directional field for the equation. Identify at least 4 isoclines. dy dx = y +x Isoclines:
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
dy/dx= y+x
![**Task:**
Sketch a directional field for the equation. Identify at least 4 isoclines.
\[ \frac{dy}{dx} = y + x \]
**Isoclines:**
(Blank space for listing isoclines)
**Directional Field:**
- A coordinate plane is provided with a cross of horizontal and vertical lines with arrows indicating standard axis directions (upward for the y-axis, rightward for the x-axis).
**Explanatory Note:**
To create a directional field for the given differential equation \(\frac{dy}{dx} = y + x\), calculate the slope \(\frac{dy}{dx}\) at various points on the plane. An isocline is a curve where a solution curve has a constant slope, so for this equation, you would set \(y + x = c\) where \(c\) is constant, to find isoclines. For example, for \(c = 0, 1, -1, 2\), the isoclines would be \(y + x = 0\), \(y + x = 1\), \(y + x = -1\), \(y + x = 2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F8fa79d17-bc14-4e39-a973-bd307d3fdb2b%2Fyvprd07_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task:**
Sketch a directional field for the equation. Identify at least 4 isoclines.
\[ \frac{dy}{dx} = y + x \]
**Isoclines:**
(Blank space for listing isoclines)
**Directional Field:**
- A coordinate plane is provided with a cross of horizontal and vertical lines with arrows indicating standard axis directions (upward for the y-axis, rightward for the x-axis).
**Explanatory Note:**
To create a directional field for the given differential equation \(\frac{dy}{dx} = y + x\), calculate the slope \(\frac{dy}{dx}\) at various points on the plane. An isocline is a curve where a solution curve has a constant slope, so for this equation, you would set \(y + x = c\) where \(c\) is constant, to find isoclines. For example, for \(c = 0, 1, -1, 2\), the isoclines would be \(y + x = 0\), \(y + x = 1\), \(y + x = -1\), \(y + x = 2\).
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