Ordinary differential equations : methods and applications

200 Pages·2008·3.207 MB·English

by  W T Ang, Y S Park

Ordinary
Differential Equations

Ordinary
Differential Equations:
Methods and Applications

W. T. Ang and Y. S. Park

Universal Publishers
Boca Raton
Ordinary Differential Equations: Methods and Applications

Copyright © 2008 W. T. Ang and Y. S. Park
All rights reserved.
No part of this book may be reproduced or transmitted in any form
or by any means, electronic or mechanical, including photocopying,
recording, or by any information storage and retrieval system,
without written permission from the publisher

Universal Publishers
Boca Raton, Florida • USA
2008

ISBN-10: 1-59942-975-6/ISBN-13: 978-1-59942-975-5 (paper)
ISBN-10: 1-59942-974-8/ISBN-13: 978-1-59942-974-8 (ebook)

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Library of Congress Cataloging-in-Publication Data

Ang, W. T., 1961-
Ordinary differential equations : methods and applications / W.T.
Ang and Y.S. Park.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-59942-975-5 (pbk. : alk. paper)
1. Differential equations. I. Park, Y. S., 1964- II. Title.
QA372.A598 2008
515'.352--dc22
2008026023
To our parents
“Everywhere, we learn from those whom
we love” Johann Wolfgang von Goethe
Preface
This introductory course in ordinary differential equations
isintendedforjuniorundergraduatestudentsinappliedmathe-
matics, science and engineering. It focuses on methods of solu-
tions and applications rather than theoretical analyses. Appli-
cations drawn mainly from dynamics, population biology and
electric circuit theory are used to show how ordinary differen-
tial equations appear in the formulation of problems in science
and engineering.
Thecalculusrequiredtocomprehendthiscourseisratherel-
ementary,involvingdifferentiation,integrationandpowerseries
representation of only real functions of one variable. A basic
knowledge of complex numbers and their arithmetic is also as-
sumed,sothatelementarycomplexfunctionswhichcanbeused
for working out easily the general solutions of certain ordinary
differentialequationscanbeintroduced. Thepre-requisitesjust
mentioned aside, the course is mainly self-contained.
The course comprises six chapters.
Chapter 1 gives the basic concepts of ordinary differential
equations, explaining what an ordinary differential equation is
and what is involved in solving such an equation. It also illus-
trates how ordinary differential equations can be derived from
physical laws or basic principles for two specific examples of
problems.
In Chapter 2, methods of solution are given for some first
order ordinary differential equations. The equations studied
include those which can be written in separable form, those
which are linear, and the nonlinear Bernoulli differential equa-
tion. Mathematical models which describe population growth
are given as examples of applications involving first order ordi-
nary differential equations.
v
InChapter3,themathematicaltheoryforconstructinggen-
eral solutions of second order linear ordinary differential equa-
tions is studied. It is applied to obtain general solutions of sec-
ond order linear ordinary differential equations with constant
coefficients and the Euler-Cauchy equations. Also discussed
is the extension of the theory to higher order linear ordinary
differential equations.
Chapter 4 shows how linear ordinary differential equations
with constant coefficients arise in the formulation of problems
involving electric circuits and spring-mass systems. Specific
examples of problems are solved.
Chapter5introducesthepowerseriesmethodandtheFrobe-
nius method for deriving series solutions of rather general ho-
mogeneous second order linear ordinary differential equations.
The methods studied can be applied to solve some well known
ordinarydifferentialequationsinmathematicalphysics,suchas
the Legendre’s equation and the Bessel’s equation, giving rise
to particular special functions, but those equations and the as-
sociated special functions are not examined in this course.
Chapter 6 describes some simple numerical methods for
solving first and second order ordinary differential equations.
For a particular example of applications, the second order non-
linear ordinary differential equation which governs the motion
of a swinging pendulum is solved numerically.
Exercises are set not only to test the understanding of stu-
dents but sometimes also to impart additional insights into the
materials studied. Suggested solutions to all the exercises are
given at the end of the chapters. To promote the use of this
course for self-study, the solutions provided are by and large
complete with details.
W. T. Ang and Y. S. Park, Singapore, 2008
vi
Contents
1 Basic concepts 10
1.1 What is an ODE? . . . . . . . . . . . . . . . 10
1.2 Solving an ODE . . . . . . . . . . . . . . . . . . 11
1.2.1 General solution . . . . . . . . . . . . . . 12
1.2.2 Particular solution . . . . . . . . . . . . . 12
1.2.3 Exact solution . . . . . . . . . . . . . . . 12
1.3 Exercise I . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Why study ODEs? . . . . . . . . . . . . . . . . . 14
1.4.1 ODE for a body in motion . . . . . . . . 14
1.4.2 ODE for a pursuit problem . . . . . . . . 18
1.5 Exercise II . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Solutions to Exercise I . . . . . . . . . . . . . . . 22
1.7 Solutions to Exercise II . . . . . . . . . . . . . . 24
2 First order ODEs 27
2.1 Preamble . . . . . . . . . . . . . . . . . . . . . . 27
2.2 First order ODEs in separable form . . . . . . . . 27
2.3 Linear 1st order ODEs . . . . . . . . . . . . . . . 33
2.3.1 Homogeneous linear 1st order ODEs . . . 34
2.3.2 Nonhomogeneous linear 1st order ODEs . 34
2.4 Bernoulli differential equation . . . . . . . . . . . 36
2.5 Population dynamics . . . . . . . . . . . . . . . . 38
2.5.1 Malthus theory of unlimited growth . . . 38
2.5.2 Verhulst theory of limited growth . . . . . 39
2.6 Exercise III . . . . . . . . . . . . . . . . . . . . . 41
2.7 Solutions to Exercise III . . . . . . . . . . . . . . 43
vii
3 Second order linear ODEs 49
3.1 Preamble . . . . . . . . . . . . . . . . . . . . . . 49
3.2 General solution of homogeneous 2nd order lin-
ear ODE . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Linearly independent functions . . . . . . 50
3.2.2 Construction of general solution . . . . . 52
3.3 Homogeneous 2nd order linear ODEs with con-
stant coefficients . . . . . . . . . . . . . . . . . . 55
3.4 Euler-Cauchy equations . . . . . . . . . . . . . . 64
3.5 Exercise IV . . . . . . . . . . . . . . . . . . . . . 70
3.6 Solving nonhomogeneous ODEs . . . . . . . . . . 71
3.6.1 Finding a particular solution by guesswork 72
3.6.2 Method of variation of parameters . . . . 78
3.7 Extension to higher order linear ODEs . . . . . . 81
3.7.1 General N-th order linear ODEs . . . . . 81
3.7.2 General solution of a homogeneous ODE . 81
3.7.3 Generalsolutionofanonhomogeneouslin-
ear ODE . . . . . . . . . . . . . . . . . . 82
3.8 Exercise V . . . . . . . . . . . . . . . . . . . . . . 83
3.9 Solutions to Exercise IV . . . . . . . . . . . . . . 85
3.10 Solutions to Exercise V . . . . . . . . . . . . . . 89
4 Circuits and springs 96
4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . 96
4.2 Electric circuits . . . . . . . . . . . . . . . . . . . 96
4.2.1 Basic electrical components . . . . . . . . 96
4.2.2 Voltage across an electric component . . . 98
4.2.3 ODEs in electric circuits . . . . . . . . . . 99
4.3 Exercise VI . . . . . . . . . . . . . . . . . . . . . 109
4.4 Spring-mass systems . . . . . . . . . . . . . . . . 109
4.4.1 A simple spring-mass system . . . . . . . 109
4.4.2 A more complicated spring-mass system . 115
4.5 Exercise VII . . . . . . . . . . . . . . . . . . . . . 118
4.6 Solutions to Exercise VI . . . . . . . . . . . . . . 119
4.7 Solutions to Exercise VII . . . . . . . . . . . . . 124
viii
5 Series solutions 128
5.1 Preamble . . . . . . . . . . . . . . . . . . . . . . 128
5.2 Review of power series . . . . . . . . . . . . . . 128
5.3 Power series method for ODEs . . . . . . . . . . 131
5.4 Exercise VIII . . . . . . . . . . . . . . . . . . . . 146
5.5 Frobenius method . . . . . . . . . . . . . . . . . 147
5.6 Exercise IX . . . . . . . . . . . . . . . . . . . . . 161
5.7 Solutions to Exercise VIII . . . . . . . . . . . . . 162
5.8 Solutions to Exercise IX . . . . . . . . . . . . . . 167
6 Numerical methods 176
6.1 Preamble . . . . . . . . . . . . . . . . . . . . . . 176
6.2 Euler’s method for 1st order ODEs . . . . . . . . 176
6.3 Second order ODEs. . . . . . . . . . . . . . . . . 183
6.4 Oscillation of a pendulum . . . . . . . . . . . . . 187
6.4.1 Nonlinear ODE . . . . . . . . . . . . . . . 187
6.4.2 ODE for ‘very small’ oscillation . . . . . . 188
6.4.3 Numerical solution for ‘larger’ oscillation 189
6.5 Numerical prudence . . . . . . . . . . . . . . . . 191
6.6 Exercise X . . . . . . . . . . . . . . . . . . . . . . 191
6.7 Solutions to Exercise X . . . . . . . . . . . . . . 194
ix
Chapter 1
Basic concepts
1.1 What is an ODE?
An equation which contains the derivative(s) of a yet to be
determined function y(x) (a function of one variable) is called
an ordinary differential equation (ODE) in y(x).
Below are some examples of ODEs in y(x):
dy
(1) 2x=0
dx −
d2y dy
(2) +3 2y(x)=x5
dx2 dx −
d3y d2y
(3) x3 +6x2 3xy(x)=sin(2x)
dx3 dx2 −
d4y d2y
(4) 2x2(y(x))10 +3x =xy(x)
dx4 dx2
An ODE in y(x) is said to be of order N if dNy/dxN is the
highest order derivative of y(x) present in the ODE.
In the examples above, (1) is an ODE of order 1 (or 1st
order ODE); (2) is of order 2; (3) is of order 3; and (4) is of
order 4.
It may be sometimes convenient to use the notation
dy d2y d3y dNy
y (x)= , y (x)= , y (x)= , , y(N)(x)= .
0 dx 00 dx2 000 dx3 ··· dxN
10