1. Consider the system x' = 3x + 2y y' = -x (a) Find the general solution of the system. (Ъ) Determine the type of the equilibrium at the origin. (c) Determine the stability of the equilibrium at the origin. (d) • in a saddle/node case, show the eigenvectors and the ray solutions. • in a center/spiral case, determine the direction of rotation. Sketch trajectories in the phase plane. Moreover,

1. Consider the system x' = 3x + 2y y' = -x (a) Find the general solution of the system. (Ъ) Determine the type of the equilibrium at the origin. (c) Determine the stability of the equilibrium at the origin. (d) • in a saddle/node case, show the eigenvectors and the ray solutions. • in a center/spiral case, determine the direction of rotation. Sketch trajectories in the phase plane. Moreover,

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

1.
Consider the system
x' = 3x + 2y
y' = -x
Find the general solution of the system.
(a)
(Ъ)
Determine the type of the equilibrium at the origin.
(c)
Determine the stability of the equilibrium at the origin.
(d)
• in a saddle/node case, show the eigenvectors and the ray solutions.
• in a center/spiral case, determine the direction of rotation.
Sketch trajectories in the phase plane. Moreover,

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The general form of a linear, homogeneous, second-order equation with constantcoefficients is (d^2 y/dt^2) + p(dy/dt) + qy = 0. (a) Write the first-order system for this equation, and write this system in matrixform.(b) Show that if q does not = 0, then the origin is the only equilibrium point of the system.(c) Show that if q does not = 0, then the only solution of the second-order equation with y constant is y(t) = 0 for all t.