Exercise 8.3.52: For each of the following nonlinear systems, i. Find all critical points (equilibria). ii. Determine the linearized system for each of the points in (i.). Classify each of the critiça points. If the critical points are not spirals or centers, find all eigenvectors. iii. Sketch the phase portrait, showing at least 4 critical points. Include the eigenvectors for the linearized system at each critical point and draw arrows on the solution curves to indicate the direction of flow. 8.3. APPLICATIONS OF NONLINEAR SYSTEMS a) x' = 2y, y' = sin x – y b) x' = -2x + 4 sin y, y' = 2x c) x' = x - y, y' = 2 sin x d) x' = 3y, y' = sin(nx) e) x' = sin(n y), y' = x + y

Exercise 8.3.52: For each of the following nonlinear systems, i. Find all critical points (equilibria). ii. Determine the linearized system for each of the points in (i.). Classify each of the critiça points. If the critical points are not spirals or centers, find all eigenvectors. iii. Sketch the phase portrait, showing at least 4 critical points. Include the eigenvectors for the linearized system at each critical point and draw arrows on the solution curves to indicate the direction of flow. 8.3. APPLICATIONS OF NONLINEAR SYSTEMS a) x' = 2y, y' = sin x – y b) x' = -2x + 4 sin y, y' = 2x c) x' = x - y, y' = 2 sin x d) x' = 3y, y' = sin(nx) e) x' = sin(n y), y' = x + y

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

Exercise 8.3.52: For each of the following nonlinear systems,
i. Find all critical points (equilibria).
ii. Determine the linearized system for each of the points in (i.). Classify each of the critiçal
points. If the critical points are not spirals or centers, find all eigenvectors.
iii. Sketch the phase portrait, showing at least 4 critical points. Include the eigenvectors for the
linearized system at each critical point and draw arrows on the solution curves to indicate the
direction of flow.
8.3. APPLICATIONS OF NONLINEAR SYSTEMS
a) x' = 2y, y' = sin x - y
%3D
b) x' = -2x + 4 sin y, y' = 2x
%3D
c) x' = x – y, y' = 2 sin x
d) x' = 3y, y' = sin(nx)
%3D
e) x' = sin(ny), y' = x + y
%3D

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