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Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? 1. y' = ? 2. y' ? 3. y' +/ = 31 1 3 -- -3 32 24. == -2 y 4 3 2 5 15 1 [1 3 121, y 2 2 1 15

Match each linear system with one of the phase plane direction fields. (The blue lines are the arrow shafts, and the black dots are the arrow tips.) ? 1. y' = ? 2. y' ? 3. y' +/ = 31 1 3 -- -3 32 24. == -2 y 4 3 2 5 15 1 [1 3 121, y 2 2 1 15

Advanced Engineering Mathematics
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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**Educational Content:** **Matching Linear Systems with Phase Plane Direction Fields** **Instructions:** Match each linear system with one of the phase plane direction fields shown on the right. The blue lines represent the arrow shafts, and the black dots depict the arrow tips. **Linear Systems:** 1. \( \vec{y}\,' = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \vec{y} \) 2. \( \vec{y}\,' = \begin{bmatrix} 1 & -3 \\ 0 & -2 \end{bmatrix} \vec{y} \) 3. \( \vec{y}\,' = \begin{bmatrix} -\frac{1}{3} & \frac{4}{5} \\ \frac{2}{5} & 1 \end{bmatrix} \vec{y} \) 4. \( \vec{y}\,' = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ 3 & 1 \end{bmatrix} \vec{y} \) **Phase Plane Direction Fields:** - **Field A:** - Shows a spiral pattern, suggesting a focus-type behavior, with trajectories moving inward toward the equilibrium point. - **Field B:** - Displays clear saddle point characteristics, with trajectories moving away from the equilibrium along certain directions and toward it along others. - **Field C:** - Features a node with direct trajectories toward the equilibrium, indicating stable node behavior. - **Field D:** - Also presents a nodal pattern but suggests an outward movement from the equilibrium point, indicative of an unstable node. **Task:** Analyze each linear system and match it with the correct phase plane direction field based on the matrix properties (eigenvalues and eigenvectors) to determine the behavior of the system.
Transcribed Image Text:**Educational Content:** **Matching Linear Systems with Phase Plane Direction Fields** **Instructions:** Match each linear system with one of the phase plane direction fields shown on the right. The blue lines represent the arrow shafts, and the black dots depict the arrow tips. **Linear Systems:** 1. \( \vec{y}\,' = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \vec{y} \) 2. \( \vec{y}\,' = \begin{bmatrix} 1 & -3 \\ 0 & -2 \end{bmatrix} \vec{y} \) 3. \( \vec{y}\,' = \begin{bmatrix} -\frac{1}{3} & \frac{4}{5} \\ \frac{2}{5} & 1 \end{bmatrix} \vec{y} \) 4. \( \vec{y}\,' = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ 3 & 1 \end{bmatrix} \vec{y} \) **Phase Plane Direction Fields:** - **Field A:** - Shows a spiral pattern, suggesting a focus-type behavior, with trajectories moving inward toward the equilibrium point. - **Field B:** - Displays clear saddle point characteristics, with trajectories moving away from the equilibrium along certain directions and toward it along others. - **Field C:** - Features a node with direct trajectories toward the equilibrium, indicating stable node behavior. - **Field D:** - Also presents a nodal pattern but suggests an outward movement from the equilibrium point, indicative of an unstable node. **Task:** Analyze each linear system and match it with the correct phase plane direction field based on the matrix properties (eigenvalues and eigenvectors) to determine the behavior of the system.
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