Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Differential Equations
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### Differential Equations and Stability Analysis #### Problem Statement: 2. A general solution of \( \mathbf{x}' = \mathbf{A}\mathbf{x} \) is given by \( \mathbf{x} = C_1 e^{-t} \begin{bmatrix} 1 \\ 2 \end{bmatrix} + C_2 e^{-2t} \begin{bmatrix} 3 \\ 1 \end{bmatrix} \). (a) Sketch the half-line solutions generated by each exponential term of the solution. Then, sketch a rough approximation of a solution in each region determined by the half-line solutions. Use arrows to indicate the direction of motion on all solutions. *Graph:* The graph provided is a coordinate plane with x-axis and y-axis ranging from -4 to 4. The goal is to sketch solution trajectories for the given differential equation, indicating the direction of motion with arrows. (b) The equilibrium at the origin is best classified as a (circle one): - Nodal sink - Nodal source - Saddle point - Spiral sink - Spiral source (*Circle one choice*). Explanation of the Graph: - The graph is a standard Cartesian coordinate system with both axes extending from -4 to 4. - The directions of the half-lines can be derived from the general solution, specifically the terms involving \( e^{-t} \) and \( e^{-2t} \). Guidance: In the analysis: 1. Determine the eigenvalues and eigenvectors of matrix \( \mathbf{A} \). 2. Use the eigenvectors to draw straight-line solutions extending from the origin. 3. Arrows should indicate whether the solutions are moving towards or away from the origin over time, based on the exponential terms in the solution: - \( e^{-t} \) indicates movement towards the origin. - \( e^{-2t} \) indicates movement towards the origin (at a faster rate than \( e^{-t} \)). For part (b), analyze the nature of the equilibrium (at the origin): - **Nodal Sink**: Solutions are straight lines approaching the origin. - **Nodal Source**: Solutions are straight lines diverging from the origin. - **Saddle Point**: Solutions diverge from the origin in some directions and converge in others. - **Spiral Sink/Source**: Solutions spiral towards/away from the origin. Using the provided solution form,
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Transcribed Image Text:### Differential Equations and Stability Analysis #### Problem Statement: 2. A general solution of \( \mathbf{x}' = \mathbf{A}\mathbf{x} \) is given by \( \mathbf{x} = C_1 e^{-t} \begin{bmatrix} 1 \\ 2 \end{bmatrix} + C_2 e^{-2t} \begin{bmatrix} 3 \\ 1 \end{bmatrix} \). (a) Sketch the half-line solutions generated by each exponential term of the solution. Then, sketch a rough approximation of a solution in each region determined by the half-line solutions. Use arrows to indicate the direction of motion on all solutions. *Graph:* The graph provided is a coordinate plane with x-axis and y-axis ranging from -4 to 4. The goal is to sketch solution trajectories for the given differential equation, indicating the direction of motion with arrows. (b) The equilibrium at the origin is best classified as a (circle one): - Nodal sink - Nodal source - Saddle point - Spiral sink - Spiral source (*Circle one choice*). Explanation of the Graph: - The graph is a standard Cartesian coordinate system with both axes extending from -4 to 4. - The directions of the half-lines can be derived from the general solution, specifically the terms involving \( e^{-t} \) and \( e^{-2t} \). Guidance: In the analysis: 1. Determine the eigenvalues and eigenvectors of matrix \( \mathbf{A} \). 2. Use the eigenvectors to draw straight-line solutions extending from the origin. 3. Arrows should indicate whether the solutions are moving towards or away from the origin over time, based on the exponential terms in the solution: - \( e^{-t} \) indicates movement towards the origin. - \( e^{-2t} \) indicates movement towards the origin (at a faster rate than \( e^{-t} \)). For part (b), analyze the nature of the equilibrium (at the origin): - **Nodal Sink**: Solutions are straight lines approaching the origin. - **Nodal Source**: Solutions are straight lines diverging from the origin. - **Saddle Point**: Solutions diverge from the origin in some directions and converge in others. - **Spiral Sink/Source**: Solutions spiral towards/away from the origin. Using the provided solution form,
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