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A dynamical system is governed by two equations: [x=y, {y = ln(x² + y)-3y. Here a dot on the top of a symbol stands for the derivative with respect to t. (a) Find critical points of this system. (b) Using linearisation of the system in the neighbourhood of each critical point, determine the nature of the critical points. (c) Draw qualitatively but neatly these critical points and corresponding trajectory diagrams.

A dynamical system is governed by two equations: [x=y, {y = ln(x² + y)-3y. Here a dot on the top of a symbol stands for the derivative with respect to t. (a) Find critical points of this system. (b) Using linearisation of the system in the neighbourhood of each critical point, determine the nature of the critical points. (c) Draw qualitatively but neatly these critical points and corresponding trajectory diagrams.

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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Question
(b) A dynamical system is governed by two equations: (a) Find critical points of this system. |x=y, [y=In(x² + y)-3y. Here a dot on the top of a symbol stands for the derivative with respect to t. (c) Using linearisation of the system in the neighbourhood of each critical point, determine the nature of the critical points. Draw qualitatively but neatly these critical points and corresponding trajectory diagrams.
Transcribed Image Text:(b) A dynamical system is governed by two equations: (a) Find critical points of this system. |x=y, [y=In(x² + y)-3y. Here a dot on the top of a symbol stands for the derivative with respect to t. (c) Using linearisation of the system in the neighbourhood of each critical point, determine the nature of the critical points. Draw qualitatively but neatly these critical points and corresponding trajectory diagrams.
Expert Solution
Step 1: Introduction

The given system of differential equations are:

table row cell x with dot on top end cell equals y row cell y with dot on top end cell equals cell ln open parentheses x cubed plus y close parentheses minus 3 y end cell end table

We need to identify the critical points.

We know that a point open parentheses a comma b close parentheses is said to be critical point of the system x with dot on top equals f open parentheses x comma y close parentheses comma space y with dot on top equals g open parentheses x comma y close parentheses if f open parentheses a comma b close parentheses equals 0 and g open parentheses a comma b close parentheses equals 0.

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