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a) Find the matrix A. Hint: A = PDP!, Note: you don't need to know A for the rest of this problem. b) Sketch the trajectories of this dynamical system, including at least one trajectory with each kind of behavior. Include all special trajectories (i.e. trajectories of eigenvectors). Trajectories should have arrows indicating direction (as k increases). c) Classify the origin as a stable or unstable equilibrium. Classify the system as a node or saddle. Alternatively, if vou prefer, vou can use the book's nomenclature: classify the origin

a) Find the matrix A. Hint: A = PDP!, Note: you don't need to know A for the rest of this problem. b) Sketch the trajectories of this dynamical system, including at least one trajectory with each kind of behavior. Include all special trajectories (i.e. trajectories of eigenvectors). Trajectories should have arrows indicating direction (as k increases). c) Classify the origin as a stable or unstable equilibrium. Classify the system as a node or saddle. Alternatively, if vou prefer, vou can use the book's nomenclature: classify the origin

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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Suppose the eigenvalues of a 2 x 2 matrix A are 3 and 5, with corresponding eigenvectors 3 = la 3 -1
Transcribed Image Text:Suppose the eigenvalues of a 2 x 2 matrix A are 3 and 5, with corresponding eigenvectors 3 = la 3 -1
a) Find the matrix A. Hint: A = PDP !. Note: you don't need to know A for the rest of this problem. b) Sketch the trajectories of this dynamical system, including at least one trajectory with each kind of behavior. Include all special trajectories (i.e. trajectories of eigenvectors). Trajectories should have arrows indicating direction (as k increases). c) Classify the origin as a stable or unstable equilibrium. Classify the system as a node or saddle. Alternatively, if you prefer, you can use the book's nomenclature: classify the origin as an attractor (stable node), repeller (unstable node), or saddle point. d) Pick a random point (not on a special trajectory) to be ro, and describe the behavior of xk as k → 00, and as k → -0o. (See parenthetical in 4.F.1.)
Transcribed Image Text:a) Find the matrix A. Hint: A = PDP !. Note: you don't need to know A for the rest of this problem. b) Sketch the trajectories of this dynamical system, including at least one trajectory with each kind of behavior. Include all special trajectories (i.e. trajectories of eigenvectors). Trajectories should have arrows indicating direction (as k increases). c) Classify the origin as a stable or unstable equilibrium. Classify the system as a node or saddle. Alternatively, if you prefer, you can use the book's nomenclature: classify the origin as an attractor (stable node), repeller (unstable node), or saddle point. d) Pick a random point (not on a special trajectory) to be ro, and describe the behavior of xk as k → 00, and as k → -0o. (See parenthetical in 4.F.1.)
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