4. Here are three vector fields, A, B, and C. Below the vector fields are some pairs of rate of change equations. Determine which of the pairs match each of the vector fields. Write an explanation for your choices.
4. Here are three vector fields, A, B, and C. Below the vector fields are some pairs of rate of change equations. Determine which of the pairs match each of the vector fields. Write an explanation for your choices.
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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![**Problem 4: Vector Fields and Rate of Change Equations**
Below are three vector fields labeled A, B, and C. Accompanying these fields are pairs of rate of change equations. Your task is to determine which pair of equations corresponds to each vector field. Provide an explanation for your choice.
**Vector Field Descriptions:**
- **Vector Field A**:
- Arrows are concentrated and circular, indicating a rotational pattern centered around the origin (0,0).
- The vector is moving counterclockwise, with the length of arrows suggesting the speed or intensity increases with distance from the center.
- **Vector Field B**:
- Arrows are more uniformly distributed across the plane.
- Shows a diagonal flow pattern from the lower left to the upper right, suggesting a directional flow without rotation.
- **Vector Field C**:
- Arrows form vertical wave-like patterns moving left to right.
- Exhibits a sinusoidal pattern with multiple peaks and troughs in the horizontal direction.
**Rate of Change Equations:**
(i) \[
\frac{dx}{dt} = x + y \\
\frac{dy}{dt} = -x + y
\]
(ii) \[
\frac{dx}{dt} = x - 0.1xy \\
\frac{dy}{dt} = -y + 0.1xy
\]
(iii) \[
\frac{dx}{dt} = 2x - 3y \\
\frac{dy}{dt} = x + y
\]
(iv) \[
\frac{dx}{dt} = x + y \\
\frac{dy}{dt} = 2x + 2y
\]
**Matching Analysis:**
1. **Field A**:
- Rotation around the center suggests a system where the derivatives include terms that cause a cyclic pattern. Equation (i) is a fit, as it has terms that rotate the flow vector around the origin.
2. **Field B**:
- The uniform diagonal flow suggests a consistent directional increase. Equation (iii), with direct proportionality and no interaction terms, fits this layout.
3. **Field C**:
- The sinusoidal pattern in Field C suggests factors akin to oscillations or dampened interactions, somewhat indicative of simple predator-prey or competitive systems. Equation (ii), which includes interaction terms that modulate growth](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff920003c-b80d-4d46-9199-437e1336b936%2Fa50cca98-5cb2-40bc-91fc-d532f17762f4%2Fgiyn80x_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Vector Fields and Rate of Change Equations**
Below are three vector fields labeled A, B, and C. Accompanying these fields are pairs of rate of change equations. Your task is to determine which pair of equations corresponds to each vector field. Provide an explanation for your choice.
**Vector Field Descriptions:**
- **Vector Field A**:
- Arrows are concentrated and circular, indicating a rotational pattern centered around the origin (0,0).
- The vector is moving counterclockwise, with the length of arrows suggesting the speed or intensity increases with distance from the center.
- **Vector Field B**:
- Arrows are more uniformly distributed across the plane.
- Shows a diagonal flow pattern from the lower left to the upper right, suggesting a directional flow without rotation.
- **Vector Field C**:
- Arrows form vertical wave-like patterns moving left to right.
- Exhibits a sinusoidal pattern with multiple peaks and troughs in the horizontal direction.
**Rate of Change Equations:**
(i) \[
\frac{dx}{dt} = x + y \\
\frac{dy}{dt} = -x + y
\]
(ii) \[
\frac{dx}{dt} = x - 0.1xy \\
\frac{dy}{dt} = -y + 0.1xy
\]
(iii) \[
\frac{dx}{dt} = 2x - 3y \\
\frac{dy}{dt} = x + y
\]
(iv) \[
\frac{dx}{dt} = x + y \\
\frac{dy}{dt} = 2x + 2y
\]
**Matching Analysis:**
1. **Field A**:
- Rotation around the center suggests a system where the derivatives include terms that cause a cyclic pattern. Equation (i) is a fit, as it has terms that rotate the flow vector around the origin.
2. **Field B**:
- The uniform diagonal flow suggests a consistent directional increase. Equation (iii), with direct proportionality and no interaction terms, fits this layout.
3. **Field C**:
- The sinusoidal pattern in Field C suggests factors akin to oscillations or dampened interactions, somewhat indicative of simple predator-prey or competitive systems. Equation (ii), which includes interaction terms that modulate growth
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