Linear Differential Equations and Oscillators PDF
Preview Linear Differential Equations and Oscillators
9780367137182_fc.indd 1 18/04/19 9:12 AM Linear Differential Equations and Oscillators © 2014 Taylor & Francis Group, LLC Mathematics and Physics for Science and Technology Series Editor: L.M.B.C. Campos Director of the Center for Aeronautical and Space Science and Technology Lisbon University Volumes in the series: Topic A – Theory of Functions and Potential Problems Volume I (Book 1) – Complex Analysis with Applications to Flows and Fields L.M.B.C. Campos Volume II (Book 2) – Elementary Transcendentals with Applications to Solids and Fluids L.M.B.C. Campos Volume III (Book 3) – Generalized Calculus with Applications to Matter and Forces L.M.B.C. Campos Topic B – Boundary and Initial-Value Problems Volume IV – Ordinary Differential Equations with Applications to Trajectories and Oscillations L.M.B.C. Campos Book 4 – Linear Differential Equations and Oscillators L.M.B.C. Campos Book 5 – Non-Linear Differential Equations and Dynamical Systems L.M.B.C. Campos Book 6 – Higher-Order Differential Equations and Elasticity L.M.B.C. Campos Book 7 – Simultaneous Differential Equations and Multi-Dimensional Vibrations L.M.B.C. Campos Book 8 – Singular Differential Equations and Special Functions L.M.B.C. Campos Book 9 – Classification and Examples of Differential Equations and their Applications L.M.B.C. Campos For more information about this series, please visit: https://www.crcpress. com/Mathematics-and-Physics-for-Science-and-Technology/book-series/ CRCMATPHYSCI 9780367137182_FM.indd 2 12/10/19 10:02 AM Mathematics and Physics for Science and Technology Volume IV Ordinary Differential Equations with Applications to Trajectories and Oscillations Book 4 Linear Differential Equations and Oscillators By L.M.B.C. Campos Director of the Center for Aeronautical and Space Science and Technology Lisbon University © 2014 Taylor & Francis Group, LLC CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-0-367-13718-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Campos, Luis Manuel Braga da Costa, author. Title: Linear differential equations and oscillators / Luis Manuel Braga da Campos. Description: Boca Raton : Taylor & Francis, a CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa, plc, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018046413| ISBN 9780367137182 (hardback : acid-free paper) | ISBN 9780429028984 (ebook) Subjects: LCSH: Vibrators–Mathematical models. | Mechanical movements– Mathematical models. | Oscillations–Mathematical models. | Differential equations, Linear. Classification: LCC TJ208 .C36 2018 | DDC 621.8–dc23 LC record available at https://lccn.loc.gov/2018046413 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com to Leonor Campos © 2014 Taylor & Francis Group, LLC Contents Contents .................................................................................................................vii Diagrams, Notes, and Tables .............................................................................xiii Series Preface .........................................................................................................xv Preface to Volume IV .........................................................................................xvii Acknowledgments ..............................................................................................xxi About the Author ..............................................................................................xxiii Mathematical Symbols ......................................................................................xxv Physical Quantities ..........................................................................................xxxv 1. Three Cases of Characteristic Polynomials ..............................................1 1.1 Equation of Order N and Initial Conditions .....................................2 1.1.1 An Ordinary Differential Equation and its Integrals .....................................................................................2 1.1.2 Arbitrary Constants in the General Integral........................3 1.1.3 Single- or Multi-Point or Mixed Boundary Values .............3 1.1.4 Example of a General Integral and of Initial Values ...........4 1.1.5 Families of Integral Curves in the Plane ..............................4 1.1.6 Regular and Singular Points of a Differential Equation....................................................................................6 1.1.7 Singular Points of Two Kinds ................................................6 1.1.8 Envelope as an Example of a Special Integral .....................9 1.1.9 Differential Equations Satisfied by a Function ..................11 1.1.10 Parametric Differential Equations and Bifurcations .............................................................................12 1.1.11 Single Solution around Stable Equilibrium .......................12 1.1.12 Three Solutions around Stable/Unstable Equilibria ........14 1.1.13 Poincaré (1892) Diagrams and Parametric Evolution .................................................................................16 1.2 General, Particular, and Complete Integrals ...................................17 1.2.1 Linear Unforced and Forced Differential Equations ........17 1.2.2 Superposition of Linearly Independent Solutions ............18 1.2.3 Wronskian (1812) and Linear Independence of Differentiable Functions .......................................................18 1.2.4 General Integral of the Linear Unforced Differential Equation (Lagrange, 1765; Fuchs, 1866) ...............................19 1.2.5 A Particular Integral of the Linear Forced Equation ..................................................................................20 1.2.6 Complete Integral of the Linear Forced Equation (D’Alembert, 1762) ..................................................................21 © 2014 Taylor & Francis Group, LLC vii viii Contents 1.3 Unforced Linear Equation with Constant Coefficients .................22 1.3.1 Distinct Roots of the Characteristic Polynomial (Euler, 1743; Cauchy, 1827) ....................................................22 1.3.2 Linear Independence of the Exponentials and Van der Monde Determinant .......................................................23 1.3.3 Two Real Distinct Roots and Hyperbolic Functions .........25 1.3.4 Complex Conjugate Roots and Circular Functions ..........27 1.3.5 Circular and Hyperbolic Functions for Fourth-Order Equations .................................................................................29 1.3.6 Root of Multiplicity M and Variation of Constants ...........30 1.3.7 Parametric Differentiation of Particular Integrals (D’Alembert, 1748) ..................................................................31 1.3.8 Multiple Roots of the Characteristic Polynomial ..............32 1.3.9 Linear Independence of Powers with Fixed Base .............33 1.3.10 Multiple Pairs of Complex Conjugate or Real Roots .........34 1.3.11 Characteristic Polynomial with Single and Multiple Roots ........................................................................................35 1.3.12 Products of Powers and Circular or Hyperbolic Functions as Particular Integrals .........................................35 1.3.13 Symmetric Real and Complex Conjugate Pairs of Roots of the Characteristic Polynomial ..............................36 1.3.14 Single and Multiple Roots of the Characteristic Polynomial ..............................................................................37 1.4 General (Complete) Integral of the Unforced (Forced) Equation ...............................................................................................38 1.4.1 Parametric Differentiation of a Forced Solution (D’Alembert, 1762) ..................................................................39 1.4.2 Rule Applied to a Forced Solution (L’Hôspital, 1696; Bernoulli, 1691) .......................................................................41 1.4.3 Resonant and Non-Resonant Forcing by an Exponential ...........................................................................43 1.4.4 Forcing by a Hyperbolic Cosine or Sine .............................44 1.4.5 Forcing by a Circular Sine or Cosine ..................................45 1.4.6 Product of an Exponential and a Hyperbolic Function ....47 1.4.7 Product of an Exponential and a Circular Function .........48 1.4.8 Product of an Exponential and a Circular and a Hyperbolic Function ..............................................................50 1.5 Polynomial of Derivatives and Inverse Operator ...........................52 1.5.1 Direct and Inverse Binomial Series .....................................53 1.5.2 Interpretation of the Inverse Polynomial of Derivatives .........................................................................54 1.5.3 Partial Fractions for the Inverse Characteristic Polynomial ..............................................................................55 1.5.4 Single and Multiple Poles of the Inverse Characteristic Polynomial ....................................................56 © 2014 Taylor & Francis Group, LLC Contents ix 1.5.5 Particular Integral for Polynomial Forcing ........................60 1.5.6 Trial Solution and Determination of Coefficients .............61 1.5.7 Series of Derivatives Applied to a Smooth Function ........62 1.5.8 Forcing by the Product of a Smooth Function and an Exponential .............................................................................64 1.5.9 Products of Smooth and Elementary Functions................65 1.5.10 Forcing by Products of Elementary Functions ..................66 1.5.11 Distinction between Resonant and Non-Resonant Cases ...............................................................................67 1.5.12 Zero as a Root of the Characteristic Polynomial ...............68 1.5.13 Forcing by the Product of a Polynomial and Exponential .............................................................................69 1.5.14 Product of a Power by a Hyperbolic Cosine or Sine .........70 1.5.15 Product of a Power by a Circular Cosine or Sine ..............71 1.5.16 Product of a Power by an Exponential and a Hyperbolic Function ..............................................................72 1.5.17 Product of a Power by an Exponential by a Circular Function...................................................................................73 1.5.18 Product of a Power by an Exponential and Circular and Hyperbolic Functions ....................................................74 1.5.19 Forced Differential Equation with Constant Coefficients .....................................................................75 1.6 Homogenous Linear Differential Equation with Power Coefficients (Euler, 1769) ....................................................................77 1.6.1 Transformation into a Linear Equation with Constant Coefficients ............................................................77 1.6.2 Simple Roots of the Characteristic Polynomial .................78 1.6.3 Multiple Roots of the Characteristic Polynomial ..............79 1.6.4 General Integral of the Homogenous Differential Equation ..................................................................................80 1.6.5 Relation between Ordinary and Homogenous Derivatives ..............................................................................81 1.6.6 Real Distinct and Complex Conjugate Roots .....................82 1.6.7 Isotropic Multidimensional Laplace Equation ..................84 1.6.8 Linear, Logarithmic, and Power-Law Potentials ...............84 1.6.9 A Second-Order Linear Homogeneous Differential Equation ..................................................................................85 1.6.10 Third-Order Linear Homogeneous Differential Equation ..................................................................................86 1.7 Homogeneous Derivatives and Characteristic Polynomial ..........86 1.7.1 Homogeneous Differential Equation Forced by a Power .......................................................................................87 1.7.2 Forcing by a Hyperbolic Cosine or Sine of a Logarithm ...............................................................................88 1.7.3 Forcing by the Circular Cosine or Sine of a Logarithm ....89 © 2014 Taylor & Francis Group, LLC
The list of books you might like
