Table Of ContentClassification and Examples
of Differential Equations
and their Applications
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
Mathematics and Physics for Science and Technology
Volume IV
Ordinary Differential
Equations with Applications
to Trajectories and Oscillations
Book 9
Classification and Examples
of Differential Equations
and their Applications
By
L.M.B.C. Campos
Director of the Center for Aeronautical
and Space Science and Technology
Lisbon University
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 334 87-2742
© 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: 978-0-367-13724-3 (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.
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
Contents
Contents .................................................................................................................vii
List of Classifications and Tables .........................................................................xi
Preface ...................................................................................................................xxi
Acknowledgments .............................................................................................xxv
About the Author .............................................................................................xxvii
Physical Quantities ...........................................................................................xxix
10. Examples 10.1 to 10.20 ....................................................................................1
EXAMPLE 10.1 One Finite Difference (Two Differential)
Equation(s) with the Same Characteristic Polynomial ....................1
E10.1.1 Linear Differential Equation with Constant
Coefficients ...............................................................................1
E10.1.2 Linear Differential Equation with Homogeneous
Coefficients ...............................................................................3
E10.1.3 Linear Finite Difference Equation with Constant
Coefficients ...............................................................................4
EXAMPLE 10.2 Energies, Dissipation, and Power of
an Oscillation .........................................................................................5
E10.2.1 Potential, Kinetic, and Total Energies ...................................5
E10.2.2 Strong Subcritical or Weakly Damped
Oscillations ...............................................................................7
E10.2.3 Averages over a Period of Energies and Power ...................8
E10.2.4 Balance of Forcing and Fast Amplification ........................10
EXAMPLE 10.3 Power-Law Forcing of a Harmonic Oscillator ...............11
E10.3.1 Power-Law Monotonic and Oscillating Forcing ...............12
E10.3.2 Forcing of an Undamped and Damped Oscillator ...........13
EXAMPLE 10.4 Five Standards of First-Order Differential
Equations ..............................................................................................15
EXAMPLE 10.5 Integrating Factor for the Linear First-Order
Differential Equation ..........................................................................17
EXAMPLE 10.6 First-Order Differentials in Two Variables .....................18
E10.6.1 Exact First-Order Differentials in Two Variables ..............19
E10.6.2 Inexact First-Order Differentials in Two Variables ...........19
E10.6.3 Integrating Factors for Inexact Differentials ......................20
EXAMPLE 10.7 Paths Near a Stagnation Point of the Second
Degree ...................................................................................................21
E10.7.1 Paths Tangent at a Stagnation Point of the
Second Degree ........................................................................21
vii
viii Contents
E10.7.2 Paths with Three Asymptotes Crossing at a
Stagnation Point .....................................................................23
E10.7.3 Comparison of Velocity Fields with a
Stagnation Point .....................................................................24
EXAMPLE 10.8 First-Order Special Differential Equations .....................25
E10.8.1 Cusped Parabola as the Envelope of a Family of
Straight Lines ..........................................................................25
E10.8.2 Smooth Parabola as the Envelope of a Family of
Straight Lines ..........................................................................26
E10.8.3 First-Order Equation of Degree Three without
Singular Integrals ..................................................................27
EXAMPLE 10.9 Differential Equations of Order Higher than
the First .................................................................................................28
E10.9.1 Linear Differential Equations with Constant or
Variable Coefficients ..............................................................29
E10.9.2 Non-Linear Second-Order Differential Equations
Omitting the Independent Variable ....................................30
E10.9.3 Higher-Order Differential Equations Reducible to
the First Order ........................................................................31
E10.9.4 Linear Non-Commutative Differential Operators
with Variable Coefficients .....................................................32
E10.9.5 Non-Linear Exact Differential Equation with
Variable Coefficients ..............................................................33
E10.9.6 Second-Order Homogeneous Differential
Equation ..................................................................................34
EXAMPLE 10.10 Bending of a Beam under Traction ................................35
E10.10.1 Elastica of a Beam under Traction .....................................36
E10.10.2 Clamped Beam with a Concentrated Torque...................37
E10.10.3 Pinned Beam with a Concentrated Force .........................40
E10.10.4 Heavy Clamped-Pinned Beam ..........................................43
E10.10.5 Cantilever Beam with a Linearly Increasing
Shear Stress .............................................................................45
EXAMPLE 10.11 Linear Bending of a Circular Heavy Plate
with a Circular Hole ...........................................................................48
EXAMPLE 10.12 Vibrations of a Membrane under Uniform or
Non-Uniform Tension ........................................................................51
E10.12.1 Propagating/Evanescent or Cut-on/Cut-off
Modes ......................................................................................54
E10.12.2 Turning Point Due to Non-Uniform Tension ...................55
EXAMPLE 10.13 Curve as the Tangent to a Vector Field
or as of the Intersection Surfaces ......................................................56
EXAMPLE 10.14 Differentials of First Degree in Three/
Four Variables .......................................................................................58
E10.14.1 Exact, Inexact, and Non-Integrable Differentials ............58
Contents ix
E10.14.2 Immediate and Complete Integrability and
Subsidiary Conditions ...........................................................59
EXAMPLE 10.15 General Boundary Conditions for the
Bending of a Plate ...............................................................................60
E10.15.1 Elastic Energy along the Boundary of a Plate..................61
E10.15.2 Integration by Parts along a Closed Regular
Boundary .................................................................................62
E10.15.3 Stress Couple and Turning Moment .................................63
E10.15.4 Plate with Arbitrary Closed Regular Boundary ..............64
E10.15.5 Isotropic and Pseudo-Isotropic Orthotropic
Plates ........................................................................................65
E10.15.6 General or Rectangular Clamped Plate ............................65
E10.15.7 Boundary Conditions for Supported or
Pinned Plates ..........................................................................66
E10.15.8 General, Circular, or Rectangular Plate
with Free Boundary ...............................................................67
E10.15.9 Sets of Boundary Conditions for the
Bending of a Plate ..................................................................67
EXAMPLE 10.16 Strong Bending of a Pseudo-Isotropic
Orthotropic Plate .................................................................................68
EXAMPLE 10.17 Non-Linear and Linear Coupled Systems of
Differential Equations .........................................................................71
E10.17.1 Simultaneous System of Non-Linear Ordinary
Differential Equations ...........................................................71
E10.17.2 Linear Differential System with a Given
Characteristic Polynomial .....................................................72
EXAMPLE 10.18 Three Coupled Systems with a Matrix of
Characteristic Polynomials ................................................................74
E10.18.1 Forced Coupled Linear Differential System
with Constant Coefficients ...................................................75
E10.18.2 Forced Coupled Linear Differential System
with Homogeneous Coefficients .........................................77
E10.18.3 Forced Coupled System of Finite Difference
Equations .................................................................................77
E10.18.4 System of Finite Difference Equations with
Oscillatory Forcing.................................................................78
E10.18.5 Forcing by Power Multiplied by a Cosine .......................78
E10.18.6 Special Case of Constant Forcing ......................................80
E10.18.7 General and Complete Solutions .......................................81
EXAMPLE 10.19 Asymptotic Stability of a Damped
Non-Linear Oscillator .........................................................................82
E10.19.1 Positive-Definite Stability Function ..................................83
E10.19.2 Negative-Definite Time Derivative of the
Stability Function ...................................................................83
