Table Of Content9780367137182_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
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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
