Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 5.2, Problem 13E
Interpretation Introduction

Interpretation:

To analyze the system of damped harmonic oscillator using second order differential equation mx¨ + bx˙ + kx = 0 where b>0 is the damping constant.

Concept Introduction:

Damped harmonic oscillator is the system in which the amplitude of oscillation decreases over time depending upon the damping factor b.

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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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