EBK NONLINEAR DYNAMICS AND CHAOS WITH S
EBK NONLINEAR DYNAMICS AND CHAOS WITH S
2nd Edition
ISBN: 9780429680151
Author: STROGATZ
Publisher: VST
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Chapter 7.6, Problem 14E
Interpretation Introduction

Interpretation:

  • (a) To derive the average equation

  • (b) To solve the averaged equation and find an approximate formula for the given differential equation for the given initial conditions.

  • (c) To solve the plot differential equation by solving it numerically and to compare it with (b)

Concept Introduction:

  • ➢ Fixed point of a differential equation is a point where f(x*) = 0 ; while substitution f(x*) = x˙ is used and x&*#x00A0;is a fixed point

  • ➢ Nullclines are the curves where either x˙=0 or y˙ = 0;they show whether the flow is completely vertical or horizontal.

  • ➢ Phase portraits represent the trajectories of the system with respect to the parameters and give a qualitative idea about evolution of the system, its fixed points, whether they will attract or repel the flow, etc.

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(b) Let I[y] be a functional of y(x) defined by [[y] = √(x²y' + 2xyy' + 2xy + y²) dr, subject to boundary conditions y(0) = 0, y(1) = 1. State the Euler-Lagrange equation for finding extreme values of I [y] for this prob- lem. Explain why the function y(x) = x is an extremal, and for this function, show that I = 2. Without doing further calculations, give the values of I for the functions y(x) = x² and y(x) = x³.
Please use mathematical induction to prove this
L sin 2x (1+ cos 3x) dx 59
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