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 Ordinary Differential Equations

Many interesting and important real life problems in the field of mathematics, physics,
chemistry, biology, engineering, economics, sociology and psychology are modelled
using the tools and techniques of ordinary differential equations (ODEs). This book
offers detailed treatment on fundamental concepts of ordinary differential equations.
Important topics including first and second order linear equations, initial value problems
and qualitative theory are presented in separate chapters. The concepts of physical models
and first order partial differential equations are discussed in detail. The text covers two-
point boundary value problems for second order linear and nonlinear equations. Using
two linearly independent solutions, a Green’s function is also constructed for given
boundary conditions.
The text emphasizes the use of calculus concepts in justification and analysis of
equations to get solutions in explicit form. While discussing first order linear systems,
tools from linear algebra are used and the importance of these tools is clearly explained
in the book. Real life applications are interspersed throughout the book. The methods
and tricks to solve numerous mathematical problems with sufficient derivations and
explanations are provided.
The first few chapters can be used for an undergraduate course on ODE, and later
chapters can be used at the graduate level. Wherever possible, the authors present the
subject in a way that students at undergraduate level can easily follow advanced topics,
such as qualitative analysis of linear and nonlinear systems.

A. K. Nandakumaran is a Professor at the Department of Mathematics, Indian Institute

of Science, Bangalore. He received the Sir C. V. Raman Young Scientist State Award in
Mathematics in 2003. His areas of interest are partial differential equations,
homogenization, control and controllability problems, inverse problems and
computations.

P. S. Datti superannuated from the Centre for Applicable Mathematics at the Tata
Institute of Fundamental Research, Bangalore after serving for over 35 years. His
research interests include nonlinear hyperbolic equations, hyperbolic conservation laws,
ordinary differential equations, evolution equations and boundary layer phenomenon.

Raju K. George is Senior Professor and Dean (R&D) at the Indian Institute of Space
Science and Technology (IIST), Thiruvananthapuram. His research areas include
functional analysis, mathematical control theory, soft computing, orbital mechanics and
industrial mathematics.
CAMBRIDGE–IISc SERIES

Cambridge–IISc Series aims to publish the best research and scholarly work on
different areas of science and technology with emphasis on cutting-edge research.
The books will be aimed at a wide audience including students, researchers,
academicians and professionals and will be published under three categories:
research monographs, centenary lectures and lecture notes.
The editorial board has been constituted with experts from a range of disciplines
in diverse fields of engineering, science and technology from the Indian Institute
of Science, Bangalore.
IISc Press Editorial Board:
G. K. Ananthasuresh, Professor, Department of Mechanical Engineering
K. Kesava Rao, Professor, Department of Chemical Engineering
Gadadhar Misra, Professor, Department of Mathematics
T. A. Abinandanan, Professor, Department of Materials Engineering
Diptiman Sen, Professor, Centre for High Energy Physics

Titles in print in this series:

• Continuum Mechanics: Foundations and Applications of Mechanics by C. S. Jog

• Fluid Mechanics: Foundations and Applications of Mechanics by C. S. Jog
• Noncommutative Mathematics for Quantum Systems by Uwe Franz and Adam
Skalski
• Mechanics, Waves and Thermodynamics by Sudhir Ranjan Jain
 Cambridge-IISc Series

Ordinary Differential Equations:

Principles and Applications

A. K. Nandakumaran
P. S. Datti
Raju K. George
University Printing House, Cambridge CB2 8BS, United Kingdom
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education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781108416412
c Cambridge University Press 2017
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2017
Printed in India
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ISBN 978-1-108-41641-2 Hardback

Additional resources for this publication at www.cambridge.org/9781108416412

Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
We would like to dedicate the book to our parents
who brought us to this wonderful world.
 Contents

Figures xiii
Preface xv
Acknowledgement xix

1 Introduction and Examples: Physical Models

1.1 A Brief General Introduction 1
1.2 Physical and Other Models 3
1.2.1 Population growth model 3
1.2.2 An atomic waste disposal problem 6
1.2.3 Mechanical vibration model 8
1.2.4 Electrical circuit 9
1.2.5 Satellite problem 10
1.2.6 Flight trajectory problem 13
1.2.7 Other examples 14
1.3 Exercises 19
1.4 Notes 21
2 Preliminaries
2.1 Introduction 22
2.2 Preliminaries from Real Analysis 22
2.2.1 Convergence and uniform convergence 22
2.3 Fixed Point Theorem 34
2.4 Some Topics in Linear Algebra 38
2.4.1 Euclidean space Rn 41
2.4.2 Points versus vectors 42
2.4.3 Linear operators 42
viii Contents

2.5 Matrix Exponential eA and its Properties 43

2.5.1 Diagonalizability and block diagonalizability 46
2.5.2 Spectral analysis of A 48
2.5.3 Computation of eJ for a Jordan block J 51
2.6 Linear Dependence and Independence of Functions 53
2.7 Exercises 54
2.8 Notes 55
3 First and Second Order Linear Equations
3.1 First Order Equations 56
3.1.1 Initial and boundary value problems 57
3.1.2 Concept of a solution 59
3.1.3 First order linear equations 60
3.1.4 Variable separable equations 65
3.2 Exact Differential Equations 66
3.3 Second Order Linear Equations 72
3.3.1 Homogeneous SLDE (HSLDE) 74
3.3.2 Linear equation with constant coefficients 77
3.3.3 Non-homogeneous equation 79
3.3.4 Green’s functions 88
3.4 Partial Differential Equations and ODE 89
3.5 Exercises 93
3.6 Notes 97
4 General Theory of Initial Value Problems
4.1 Introduction 99
4.1.1 Well-posed problems 99
4.1.2 Examples 100
4.2 Sufficient Condition for Uniqueness of Solution 103
4.2.1 A basic lemma 103
4.2.2 Uniqueness theorem 106
4.3 Sufficient Condition for Existence of Solution 107
4.3.1 Cauchy–Peano existence theorem 112
4.3.2 Existence and uniqueness by fixed point theorem 116
4.4 Continuous Dependence of the Solution on
Initial Data and Dynamics 119
 Contents ix

4.5 Continuation of a Solution into Larger Intervals and Maximal

Interval of Existence 120
4.5.1 Continuation of the solution outside the
interval |t − t0 | ≤ h 121
4.5.2 Maximal interval of existence 123
4.6 Existence and Uniqueness of a System of Equations 125
4.6.1 Existence and uniqueness results for systems 127
4.7 Exercises 130
4.8 Notes 133
5 Linear Systems and Qualitative Analysis
5.1 General nth Order Equations and Linear Systems 134
5.2 Autonomous Homogeneous Systems 136
5.2.1 Computation of etA in special cases 137
5.3 Two-dimensional Systems 139
5.3.1 Computation of eB j and etB j 140
5.4 Stability Analysis 143
5.4.1 Phase plane and phase portrait 143
5.4.2 Dynamical system, flow, vector fields 145
5.4.3 Equilibrium points and stability 147
5.5 Higher Dimensional Systems 155
5.6 Invariant Subspaces under the Flow etA 165
5.7 Non-homogeneous, Autonomous Systems 167
5.7.1 Solution to non-homogeneous systems
(variation of parameters) 168
5.7.2 Non-autonomous systems 169
5.8 Exercises 175
5.9 Notes 179
6 Series Solutions: Frobenius Theory
6.1 Introduction 180
6.2 Real Analytic Functions 180
6.3 Equations with Analytic Coefficients 183
6.4 Regular Singular Points 189
6.4.1 Equations with regular singular points 190
6.5 Exercises 198
6.6 Notes 200
x Contents

7 Regular Sturm–Liouville Theory

7.1 Introduction 201
7.2 Basic Result and Orthogonality 203
7.3 Oscillation Results 208
7.3.1 Comparison theorems 211
7.3.2 Location of zeros 218
7.4 Existence of Eigenfunctions 219
7.5 Exercises 220
7.6 Notes 222
8 Qualitative Theory
8.1 Introduction 223
8.2 General Definitions and Results 224
8.2.1 Examples 228
8.3 Liapunov Stability, Liapunov Function 229
8.3.1 Linearization 230
8.3.2 Examples 233
8.4 Liapunov Function 238
8.5 Invariant Subspaces and Manifolds 245
8.6 Phase Plane Analysis 248
8.6.1 Examples 248
8.7 Periodic Orbits 253
8.8 Exercises 264
8.9 Notes 266
9 Two Point Boundary Value Problems
9.1 Introduction 267
9.2 Linear Problems 269
9.2.1 BVP for linear systems 274
9.2.2 Examples 275
9.3 General Second Order Equations 276
9.3.1 Examples 280
9.4 Exercises 284
9.5 Notes 284
10 First Order Partial Differential Equations:
Method of Characteristics
10.1 Linear Equations 286
 Contents xi

10.2 Quasi-linear Equations 290

10.3 General First Order Equation in Two Variables 295
10.4 Hamilton–Jacobi Equation 302
10.5 Exercises 304
10.6 Notes 306
Appendix A Poincarè–Bendixon and Leinard’s Theorems
A.1 Introduction 307
A.2 Poincarè–Bendixon Theorems 310
A.2.1 Intersection with transversals 310
A.3 Leinard’s Theorem 316
Bibliography 321
Index 325
 Figures

1.1 Logistic map 5

1.2 A basic LCR circuit 9
1.3 Satellite problem 11
1.4 Flight trajectory 13
3.1 Free undamped vibrations 84
3.2 Free damped vibration 85
3.3 Forced undamped vibrations with resonance 87
3.4 (a) Characteristic curves, (b) Solution curves 90
3.5 Characteristic and initial curves 91
4.1 Picard’s theorem 108
4.2 Cauchy–Peano existence 113
5.1 Saddle point equilibrium 145
5.2 Vector field 146
5.3 (a) Unstable node, (b) Stable node 148
5.4 (a) Stable node, (b) Unstable node 149
5.5 (a) Stable node, (b) Unstable node 150
5.6 Centre 151
5.7 Stable focus 152
5.8 Unstable focus 152
5.9 Degenerate case 153
5.10 Bifurcation diagram 155
xiv Figures

5.11 Phase portrait of a 3 × 3 system 156

5.12 Phase portrait of a 3 × 3 system 158
8.1 Phase line for ẋ = sin x 233
8.2 Proof of Liapunov’s theorem 240
8.3 Domain for the proof of Chetaev’s theorem 242
8.4 Potential function and phase plane for the pendulum equation 249
8.5 Potential function and phase portrait of Duffing oscillator 252
8.6 A typical closed curve Γ with a vector field 254
8.7 The closed curve to prove Iv (Γ1 ) = Iv (Γ2 ) 257
8.8 The closed curve to prove Theorem 8.7.7 259
8.9 Proof of Theorem 8.7 259
9.1 Phase portrait for ü = 2 + u2 280
9.2 Phase portrait for ü + λ eu = 0, λ > 0 281
9.3 Phase portrait for ü − λ eu = 0, λ > 0, E ≥ 0 282
9.4 Phase portrait for ü − λ eu = 0, λ > 0, E < 0 283
10.1 A characteristic curve 287
10.2 Characteristic and initial curves 292
10.3 Monge cone 295
10.4 Characteristic strips 298
10.5 Characteristic and initial strips 300
A.1 Intersection of orbits with a transversal 311
A.2 The direction of an orbit while crossing a transversal 313
A.3 Crossings of an orbit on a transversal 313
A.4 Proof of Leinard’s theorem 318
 Preface

Many interesting and important real life problems are modeled using
ordinary differential equations (ODE). These include, but are not limited
to, physics, chemistry, biology, engineering, economics, sociology,
psychology etc. In mathematics, ODE have a deep connection with
geometry, among other branches. In many of these situations, we are
interested in understanding the future, given the present phenomenon. In
other words, we wish to understand the time evolution or the dynamics of
a given phenomenon. The subject field of ODE has developed, over the
years, to answer adequately such questions. Yet, there are many
important intriguing situations, where complete answers are still awaited.
The present book aims at giving a good foundation for a beginner,
starting at an undergraduate level, without compromising on the rigour.
We have had several occasions to teach the students at the
undergraduate and graduate level in various universities and institutions
across the country, including our own institutions, on many topics
covered in the book. In our experience and the interactions we have had
with the students, we felt that many students lack a clear notion of ODE
including the simplest integral calculus problem. For other students, a
course on ODE meant learning a few tricks to solve equations. In India,
in particular, the books which are generally prescribed, consist of a few
tricks to solve problems, making ODE one of the most uninteresting
subject in the mathematical curriculum. We are of the opinion that many
students at the beginning level do not have clarity about the essence of
ODE, compared to other subjects in mathematics.
While we were still contemplating to write a book on ODE, to address
some of the issues discussed earlier, we got an opportunity to present
a video course on ODE, under the auspices of the National Programme
xvi Preface

for Technology Enhanced Learning (NPTEL), Department of Science

and Technology (DST), Government of India, and our course is freely
available on the NPTEL website (see www.nptel.ac.in/courses/
111108081). In this video course, we have presented several topics. We
have also tried to address many of the doubts that students may have at
the beginning level and the misconceptions some other students may
possess.
Many in the academic fraternity, who watched our video course,
suggested that we write a book. Of course, writing a text book, that too
about a classical subject at a beginning level, meant a much bigger task
than a video course, involving choosing and presenting the material in a
very systematic way. In a way, the video course may supplement the book
as it gives a flavour of a classroom lecture. We hope that in this way,
students in remote areas and/or places where there is lack of qualified
teachers, benefit from the book and the video course, making good use of
the modern technology available through the Internet. The teachers of
undergraduate courses can also benefit, we hope, from this book in fine
tuning their skills in ODE.
We have written the present book with the hope that it can also be used
at the undergraduate level in universities everywhere, especially in the
context of Indian universities, with appropriately chosen topics in
Chapters 1, 2 and 3. As the students get more acquainted with basic
analysis and linear algebra, the book can be introduced at the graduate
level as well and even at the beginning level of a research programme.
We now briefly describe the contents of the book. The book has a total
of ten chapters and one appendix.
Chapter 1 describes some important examples from real life situations
in the field of physics to biology to engineering. We thought this as a
very good motivation for a beginner to undertake the study of ODE; in a
rigorous course on ODE, often a student does not see a good reason to
study the subject. We have observed that this has been one of the major
concerns faced by students at a beginning level.
As far as possible, we have kept the prerequisite to a minimum: a good
course on calculus. With this in mind, we have collected, in Chapter 2,
a number of important results from analysis and linear algebra that are
used in the main text. Wherever possible, we have provided proofs and
simple presentations. This makes the book more or less self contained,
though a deeper knowledge in analysis and linear algebra will enhance the
understanding of the subject.
 Preface xvii

First and second order equations are dealt with in Chapter 3. This
chapter also contains the usual methods of solutions, but with sufficient
mathematical explanation, so that students feel that there is indeed
rigorous mathematics behind these methods. The concept behind the
exact differential equation is also explained. Second order linear
equations, with or without constant coefficients, are given a detailed
treatment. This will make a student better equipped to study linear
systems, which are treated in Chapter 5.
Chapter 4 deals with the hard theme of existence, non-existence,
uniqueness etc., for a single equation and also a system of first order
equations. We have tried to motivate the reader to wonder why these
questions are important and how to deal with them. We have also
discussed other topics such as continuous dependence on initial data,
continuation of solutions and the maximal interval of existence of a
solution.
Linear systems are studied in great detail in Chapter 5. We have tried to
show the power of linear algebra in obtaining the phase portrait of 2 × 2
and general systems. We have also included a brief discussion on Floquet
theory, which deals with linear systems with periodic coefficients.
In the case of a second order linear equation with variable coefficients,
it is not possible in general, to obtain a solution in explicit form. This has
been discussed at length in Chapter 3. Chapter 6 deals with a class of
second order linear equations, whose solutions may be written explicitly,
although in the form of an infinite series. This method is attributed to
Frobenius.
Chapter 7 deals with the regular Sturm–Lioville theory. This theory is
concerned with boundary value problems associated with linear second
order equations with smooth coefficients, in a compact interval on the
real, involving a parameter. We, then, show the existence of a countable
number of values of the parameter and associated non-trivial solutions of
the differential equation satisfying the boundary conditions. There are
many similarities with the existence of eigenvalues and eigenvectors of a
matrix, though we are now in an infinite dimensional situation.
The qualitative theory of nonlinear systems is the subject of Chapter 8.
The contents may be suitable for a senior undergraduate course or a
beginning graduate course. This chapter does demand for more
prerequisites and these are described in Chapter 2. The main topics of the
chapter are equilibrium points or solutions of autonomous systems and
their stability analysis; existence of periodic orbits in a two-dimensional
xviii Preface

system. We have tried to make a presentation of these important notions

so that it can be easily understood by any student at a senior
undergraduate level. The proofs of two important theorems on the
existence of periodic orbits are given in the Appendix.
Chapter 9 considers the study of two point boundary value problems
for second order linear and nonlinear equations. The first dealing with
linear equations fully utilises the theory developed in Chapter 3. Using
two linearly independent solutions, a Green’s function is constructed for
given boundary conditions. This is similar to an integral calculus
problem. For nonlinear equations, we no longer have the luxury of two
linearly independent solutions. A result which gives a taste of delicate
analysis is proved. It is also seen through some examples how phase
plane analysis can help in deciding whether a given boundary value
problem has a solution or not.
In Chapter 10, we have attempted to show how the methods of ODE are
used to find solutions of first order partial differential equations (PDE). We
essentially describe the method of characteristics for solving general first
order PDE. As very few books on ODE deal with this topic, we felt like
including this, as a student gets some benefit of studying PDE and (s)he
can later pursue a course on PDE.
We have followed the standard notations. Vectors in Euclidean faces
and matrices are in boldface.
 Acknowledgement

We wish to express our sincere appreciation to Gadadhar Misra and

others at the IISc Press for suggesting to publish our book through the
joint venture of IISc Press and Cambridge University Press. We also
would like to thank Gadadhar Misra for all the help in this regard. We
wish to acknowledge the support we received from our respective
institutions and the moral support from our colleagues, during the
preparation of the manuscript. We thank our academic fraternity, who
have made valuable suggestions after reading through the various parts of
the book. We would like to thank the students who attended our lectures
at various places and contributed in a positive way. Over the years, we
have had the opportunity to deliver talks in various lecture programs
conducted by National Programme in Differential Equations (NPDE),
India and the Indian Science Acadamies; our sincere thanks to them. We
also wish to thank the anonymous referees for their constructive criticism
and suggestions, which have helped us in improving the presentation.
The illustrations have been drawn using the freely available software
packages tikz and circuitikz. We are also thankful to the CUP team for
their coordination from the beginning and their excellent production. Last
but not the least, we wish to thank our family members for their patience
and support during the preparation of this book.
1
Introduction and Examples:
Physical Models

1.1 A Brief General Introduction

The beginning of the study of ordinary differential equations (ODE)
could perhaps be attributed to Newton and Leibnitz, the inventors of
differential and integral calculus. The theory began in the late 17th
century with the early works of Newton, Leibnitz and Bernoulli. As was
customary then, they were looking at the fundamental problems in
geometry and celestial mechanics. There were also important
contributions to the development of ODE, in the initial stages, by great
mathematicians – Euler, Lagrange, Laplace, Fourier, Gauss, Abel,
Hamilton and others. As the modern concept of function and analysis
were not developed at that time, the aim was to obtain solutions of
differential equations (and in turn, solutions to physical problems) in
terms of elementary functions. The earlier methods in this direction are
the concepts of integrating factors and method of separation of variables.
In the process of developing more systematic procedures, Euler,
Lagrange, Laplace and others soon realized that it is hopeless to discover
methods to solve differential equations. Even now, there are only a
handful of sets of differential equations, that too in a simpler form, whose
solutions may be written down in explicit form. It is in this scenario that
the qualitative analysis – existence, uniqueness, stability properties,
asymptotic behaviour and so on – of differential equations became very
important. This qualitative analysis depends on the development of other
branches of mathematics, especially analysis. Thus, a second phase in the
study of differential equations started from the beginning of the 19th
century based on a more rigorous approach to calculus via the
2 Ordinary Differential Equations: Principles and Applications

mathematical analysis. We remark that the first existence theorem for first
order differential equations is due to Cauchy in 1820. A class of
differential equations known as linear differential equations, is much
easier to handle. We will analyse linear equations and linear systems in
more detail and see the extensive use of linear algebra; in particular, we
will see how the nature of eigenvalues of a given matrix influences the
stability of solutions.
After the invention of differential calculus, the question of the
existence of antiderivative led to the following question regarding
differential equation: Given a function f , does there exist a function g
such that ġ(t ) = f (t )? Here, ġ(t ) is the derivative of g with respect to t.
This was the beginning of integral calculus and we refer to this problem
as an integral calculus problem. In fact, Newton’s second law of motion
describing the motion of a particle having mass m states that the rate
change of momentum equals the applied force. Mathematically, this is
written as dtd (mv) = −F, where v is the velocity of the particle. If
x = x(t ) is the position of the particle at time t, then v(t ) = ẋ(t ). In
general, the applied force F is a function of t, x and v. If we assume F is
a function of t, x, we have a second order equation for x given by
mẍ = −F (t, x). If F is a function of x alone, we obtain a conservative
equation which we study in Chapter 8. If on the other hand, F is a
function of t alone, then the second law leads to two integral calculus
problems: namely, first solve for the momentum p = mv by ṗ = −F (t )
and then solve for the position using mẋ = p. This also suggests that one
of the best ways to look at a differential equation is to view it as a
dynamical system; namely, the motion of some physical object. Here t,
the independent variable is viewed as time and x is the unknown variable
which depends on the independent variable t, and is known as the
dependent variable.
A large number of physical and biological phenomena can be
modelled via differential equations. Applications arise in almost all
branches of science and engineering–radiation decay, aging, tumor
growth, population growth, electrical circuits, mechanical vibrations,
simple pendulum, motion of artificial satellites, to mention a few.
In summary, real life phenomena together with physical and other
relevant laws, observations and experiments lead to mathematical models
(which could be ODE). One would like to do mathematical analysis and
computations of solutions of these models to simulate the behaviour of
these physical phenomena for better understanding.
 Introduction and Examples: Physical Models 3

Definition 1.1.1
An ODE is an equation consisting of an independent variable t, an
unknown function (dependent variable) y = y(t ) and its derivatives up
to a certain order. Such a relation can be written as
dny
 
dy
f t, y, , · · · , n = 0. (1.1.1)
dt dt
Here, n is a positive integer, known as the order of the differential
equation.

For example, first and second order equations, respectively, can be written
as
dy d 2 y
   
dy
f t, y, = 0 and f t, y, , 2 = 0. (1.1.2)
dt dt dt
We will be discussing some special cases of these two classes of
equations. It is possible that there will be more than one unknown
function and in that case, we will have a system of differential equations.
A higher order differential equation in one unknown function may be
reduced into a system of first order differential equations. On the other
hand, if there are more than one independent variable, we end up with
partial differential equations (PDEs).

1.2 Physical and Other Models

We begin with a few mathematical models of some real life problems and
present solutions to some of these problems. However, methods of
obtaining such solutions will be introduced in Chapter 3, and so are the
terminologies like linear and nonlinear equations.

1.2.1 Population growth model

We begin with a linear model. If y = y(t ), represents the population size
dy
of a given species at time t, then the rate of change of population is
dt
proportional to y(t ) if there is no other species to influence it and there is
no net migration. Thus, we have a simple linear model [Bra78]
dy
= ry(t ), (1.2.1)
dt
4 Ordinary Differential Equations: Principles and Applications

where r denotes the difference between birth rate and death rate. If y(t0 ) =
y0 is the population at time t0 , our problem is to find the population for all
t > t0 . This leads to the so-called initial value problem (IVP) which will
be discussed in Chapter 3. Assuming that r is a constant, the solution is
given by

y(t ) = y0 er(t−t0 ) (1.2.2)

Note that, if r > 0, then as t → ∞, the population y(t ) → ∞. Indeed, this
linear model is found to be accurate when the population is small and for
small time. But it cannot be a good model as no population, in reality, can
grow indefinitely. As and when the population becomes large, there will
be competition among the population entities for the limited resources like
food, space etc.
This suggests that we look for a more realistic model which is given
by the following logistic nonlinear model. The statistical average of the
number of encounters of two members per unit time is proportional to y2 .
Thus, a better model would be
dy
= ay − by2 , y(t0 ) = y0 . (1.2.3)
dt
Here a, b are positive constants. The negative sign in the quadratic term
represents the competition and reduces the growth rate. This is known
as the logistic law of population growth. It was introduced by the Dutch
mathematical biologist Verhulst in 1837. It is also known as the Malthus
law.
Practically, b is small compared to a. Thus, if y is not too large, then
2
by will be negligible compared to ay and the model behaves similar to
the linear model. However, when y becomes large, the term by2 will have
a considerable influence on the growth of y, as can be seen from the
following discussion.
The solution of (1.2.3) is given by1
1 |y| a − by0
log = t − t0 , t > t0 . (1.2.4)
a |y0 | a − by
Note that y ≡ 0 and y ≡ ab are solutions to the nonlinear differential
equation in (1.2.3) with the initial condition y(t0 ) = 0 and y(t0 ) = ab ,
1 The reader, after getting familiarised with the methods of solutions in Chapter 3, should work out

the details for this and the other examples in this chapter.
 Introduction and Examples: Physical Models 5

respectively. Hence, if the initial population y0 satisfies 0 < y0 < ba , then

the solution will remain in the same interval for all time. This follows
from the existence and uniqueness theory, which will be developed in
Chapter 4. A simplification of (1.2.4) gives
ay0
y(t ) = . (1.2.5)
by0 + (a − by0 )e−a(t−t0 )

a/b

a/2b

Fig. 1.1 Logistic map

a
In case 0 < y0 < , the curve y(t ) is depicted as in Fig. 1.1. This curve is
b
called the logistic curve; it is also called an S-shaped curve, because of its
a
shape. Note that is the limiting population, also known as capacity of
b
dy
the ecological environment. In this case, the rate of population is
dt
d2y dy
positive and hence, y is an increasing function. Since 2 = (a − 2by) ,
dt dt
we immediately see that it is positive if the population is between 0 and
a
half the limiting population, namely, , whereas, it is negative when the
2b
6 Ordinary Differential Equations: Principles and Applications

a
population crosses the half way mark . This indicates that if the initial
2b
population is less than half the limiting
 population, then there is an
2

dy d y
accelerated growth > 0, 2 > 0 , but after reaching half the
dt dt  
dy
population, the population still grows > 0 , but it has now a
 2  dt
d y
decelerated growth <0 .
dt 2
When we analyse the case where the initial population is bigger than
dy d2y
the limiting population, we observe that < 0 and 2 < 0. Thus, the
dt dt
population decreases with a decelerated growth to the limiting population.

Remark 1.2.1

The estimation of the vital coefficients a and b in a particular

population model is indeed an important issue which has to be
updated in a period of time as they are influenced by other parameters
like pollution, sociological trends, etc. In a more realistic model, one
needs to consider more than one species, their interactions,
unforeseen issues like epidemics, natural disasters, etc., which may
lead to more complicated equations.

1.2.2 An atomic waste disposal problem

The dumping of tightly sealed drums containing highly concentrated
radioactive waste in the sea below a certain depth (say 300 feet) from the
surface is a very sensitive issue as it could be environmentally hazardous.
The drums could break due to the impact of their velocity exceeding a
certain limit, say 40 ft/sec. Our problem is to compute the velocity by
using Newton’s second law of motion and assess the level of safety
involved in the process. Let y(t ) denote the position, at time t, of the
object, the drum, (considered as a particle) measured from the sea surface
(indicating y = 0) as a positive quantity. The total force acting on the
object is given by
F = W − B − D,
where the weight W = mg is the force due to gravity, B is the buoyancy
force of water acting against the forward movement and D = cV is the drag
 Introduction and Examples: Physical Models 7

dy
exerted by water (it is a kind of resistance), where V = , the velocity
dt
of the object and c > 0 is a constant of proportionality. Thus, we have the
differential equation
d2y 1 1 g
2
= F = (W −B−cV ) = (W −B−cV ), y(0) = 0. (1.2.6)
dt m m W
Equivalently,
dV cg g
+ V = (W − B), V (0) = 0. (1.2.7)
dt W W
Equation (1.2.7) can be solved to get
W −B  cg

V (t ) = 1 − e− W t . (1.2.8)
c
Thus, V (t ) is increasing and tends to W −B c as t → ∞ and the value
W −B
(practically) of ≈ 700.
c
The limiting value 700 ft/sec of velocity is far above the permitted
critical value. Thus, it remains to ensure that V (t ) does not reach 40 ft/sec
by the time it reaches the sea bed. But it is not possible to compute t at
which time the drum hits the sea bed and one needs to do further analysis.

Analysis: The idea is to view the velocity V (t ) not as a function of

time, but as a function of position y. Let v(y) be the velocity at height y
measured from the surface of the sea downwards. Then, clearly,
dV dv dy dv
V (t ) = v(y(t )) so that = = v . Hence, (1.2.7) becomes
dt dy dt dy

v dv g

 = ,
W − B − cv dy W (1.2.9)

v(0) = 0.


This is a first order non-homogeneous nonlinear equation for the velocity

v. Indeed, the equation is more difficult, but it is in a variable separable
form and can be integrated easily. We can solve this equation to obtain the
solution in the form
gy v W −B W − B − cv
= − − 2 log . (1.2.10)
W c c W −B
8 Ordinary Differential Equations: Principles and Applications

Of course, v cannot be explicitly expressed in terms of y as it is a

nonlinear equation. However, it is possible to obtain accurate estimates
for the velocity v(y) at height y and it is estimated that v(300) ≈ 45 ft/sec
and hence, the drum could break at a depth of 300 feet.

Tail to the Tale: This problem was initiated when environmentalists

and scientists questioned the practice of dumping waste materials by the
Atomic Energy Commission of USA. After the study, the dumping of
atomic waste was forbidden, in regions of sea not having sufficient
depths.

1.2.3 Mechanical vibration model

The fundamental mechanical model, namely spring-mass-dashpot system
(SMD) has applications in shock absorbers in automobiles, heavy guns,
etc. An object of mass m is attached to an elastic spring of length l which
is suspended from a rigid horizontal body. This is a spring–mass system.
Elastic spring has the property that when it is stretched or compressed
by a small length ∆l, it will exert a force of magnitude proportional to
∆l, say k∆l in the opposite direction of stretching or compressing. The
positive constant k is called spring constant which is a measure of stiffness
of the spring. We then obtain an SMD system when this spring–mass is
immersed in a medium like oil which will also resist the motion of the
spring–mass. In a simple situation, we may assume that the force exerted
by the medium on the spring–mass is proportional to the velocity of the
mass and in the opposite direction of the movement of mass. It is also
similar to a seismic instrument used to obtain a seismograph to detect the
motion of the earth’s surface.
Let y(t ) denote the position of mass at time t, y = 0 being the position
of the mass at equilibrium and let us take the downward direction as
positive. There are four forces acting on the system, that is, F = W + R+
D + F0 , where W = mg, the force due to gravity; R = −k(∆l + y), the
restoring force; D, the damping or drag force and F0 , the external applied
force, if any. Drag force is the kind of resistance force which the medium
exerts on the mass and hence, it will be negative. It is usually
dy
proportional to the velocity, that is, D = −c . At equilibrium, the
dt
spring has been stretched a length ∆l and so k∆l = mg. Applying
Newton’s second law, we get
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The Project Gutenberg eBook of The house at
Pooh Corner
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Title: The house at Pooh Corner

Author: A. A. Milne

Illustrator: Ernest H. Shepard

Release date: February 21, 2024 [eBook #73011]

Language: English

Original publication: New York, NY: E.P. Dutton & Co., Inc, 1928

Credits: Greg Weeks, Mary Meehan and the Online Distributed

Proofreading Team at http://www.pgdp.net

*** START OF THE PROJECT GUTENBERG EBOOK THE HOUSE

AT POOH CORNER ***
THE HOUSE AT POOH CORNER

BY A. A. MILNE

with decorations
by Ernest H. Shepard

PUBLISHED BY
E. P. DUTTON & CO., INC., NEW YORK

THE HOUSE AT POOH CORNER

COPYRIGHT, 1928, BY E. P. DUTTON & CO., INC.

ALL RIGHTS RESERVED
PRINTED IN U. S. A.

First Printing September, 1928

100th Printing December, 1936

 139th Printing July, 1949

Reprinted, from new plates and engravings

and type entirely reset August, 1950

141st Printing September, 1951

PRINTED IN THE UNITED STATES OF AMERICA BY THE

AMERICAN BOOK-STRATFORD PRESS, INC., NEW YORK
 DEDICATION

You gave me Christopher Robin, and then

You breathed new life in Pooh.
Whatever of each has left my pen
Goes homing back to you.
My book is ready, and comes to greet
The mother it longs to see—
It would be my present to you, my sweet,
If it weren't your gift to me.
 Contradiction
An introduction is to introduce people, but Christopher Robin and his
friends, who have already been introduced to you, are now going to
say Good-bye. So this is the opposite. When we asked Pooh what
the opposite of an Introduction was, he said "The what of a what?"
which didn't help us as much as we had hoped, but luckily Owl kept
his head and told us that the opposite of an Introduction, my dear
Pooh, was a Contradiction; and, as he is very good at long words, I
am sure that that's what it is.
Why we are having a Contradiction is because last week when
Christopher Robin said to me, "What about that story you were going
to tell me about what happened to Pooh when——" I happened to
say very quickly, "What about nine times a hundred and seven?" And
when we had done that one, we had one about cows going through a
gate at two a minute, and there are three hundred in the field, so how
many are left after an hour and a half? We find these very exciting,
and when we have been excited quite enough, we curl up and go to
sleep ... and Pooh, sitting wakeful a little longer on his chair by our
pillow, thinks Grand Thoughts to himself about Nothing, until he, too,
closes his eyes and nods his head, and follows us on tip-toe into the
Forest. There, still, we have magic adventures, more wonderful than
any I have told you about; but now, when we wake up in the morning,
they are gone before we can catch hold of them. How did the last one
begin? "One day when Pooh was walking in the Forest, there were
one hundred and seven cows on a gate...." No, you see, we have lost
it. It was the best, I think. Well, here are some of the other ones, all
that we shall remember now. But, of course, it isn't really Good-bye,
because the Forest will always be there ... and anybody who is
Friendly with Bears can find it.
A. A. M.
 Contents
I. IN WHICH A House Is Built at Pooh Corner for Eeyore
II. IN WHICH Tigger Comes to the Forest and Has Breakfast
IN WHICH A Search Is Organized, and Piglet Nearly Meets the
III.
Heffalump Again
IV. IN WHICH It Is Shown That Tiggers Don't Climb Trees
IN WHICH Rabbit Has a Busy Day, and We Learn What
V.
Christopher Robin Does in the Mornings
VI. IN WHICH Pooh Invents a New Game and Eeyore Joins In
VII. IN WHICH Tigger Is Unbounced
VIII. IN WHICH Piglet Does a Very Grand Thing
IX. IN WHICH Eeyore Finds the Wolery and Owl Moves Into It
IN WHICH Christopher Robin and Pooh Come to an Enchanted
X.
Place, and We Leave Them There
 THE HOUSE AT POOH CORNER

CHAPTER I
IN WHICH A House Is Built at Pooh Corner for Eeyore
One day when Pooh Bear had nothing else to do, he thought he
would do something, so he went round to Piglet's house to see what
Piglet was doing. It was still snowing as he stumped over the white
forest track, and he expected to find Piglet warming his toes in front
of his fire, but to his surprise he saw that the door was open, and the
more he looked inside the more Piglet wasn't there.
"He's out," said Pooh sadly. "That's what it is. He's not in. I shall have
to go a fast Thinking Walk by myself. Bother!"
But first he thought that he would knock very loudly just to make quite
sure ... and while he waited for Piglet not to answer, he jumped up
and down to keep warm, and a hum came suddenly into his head,
which seemed to him a Good Hum, such as is Hummed Hopefully to
Others.

The more it snows

(Tiddely pom),
The more it goes
(Tiddely pom),
The more it goes
(Tiddely pom),
On snowing.
And nobody knows
(Tiddely pom),
How cold my toes
 (Tiddely pom),
How cold my toes
(Tiddely pom),
Are growing.

"So what I'll do," said Pooh, "is I'll do this. I'll just go home first and
see what the time is, and perhaps I'll put a muffler round my neck,
and then I'll go and see Eeyore and sing it to him."
He hurried back to his own house; and his mind was so busy on the
way with the hum that he was getting ready for Eeyore that, when he
suddenly saw Piglet sitting in his best arm-chair, he could only stand
there rubbing his head and wondering whose house he was in.
"Hallo, Piglet," he said. "I thought you were out."
"No," said Piglet, "it's you who were out, Pooh."
"So it was," said Pooh. "I knew one of us was."
He looked up at his clock, which had stopped at five minutes to
eleven some weeks ago.
"Nearly eleven o'clock," said Pooh happily. "You're just in time for a
little smackerel of something," and he put his head into the cupboard.
"And then we'll go out, Piglet, and sing my song to Eeyore."
"Which song, Pooh?"
"The one we're going to sing to Eeyore," explained Pooh.
The clock was still saying five minutes to eleven when Pooh and
Piglet set out on their way half an hour later. The wind had dropped,
and the snow, tired of rushing round in circles trying to catch itself up,
now fluttered gently down until it found a place on which to rest, and
sometimes the place was Pooh's nose and sometimes it wasn't, and
in a little while Piglet was wearing a white muffler round his neck and
feeling more snowy behind the ears than he had ever felt before.
"Pooh," he said at last, and a little timidly, because he didn't want
Pooh to think he was Giving In, "I was just wondering. How would it
be if we went home now and practised your song, and then sang it to
Eeyore tomorrow—or—or the next day, when we happen to see
him?"
"That's a very good idea, Piglet," said Pooh. "We'll practise it now as
we go along. But it's no good going home to practise it, because it's a
special Outdoor Song which Has To Be Sung In The Snow."
"Are you sure?" asked Piglet anxiously.
"Well, you'll see, Piglet, when you listen. Because this is how it
begins. The more it snows, tiddely pom——"
"Tiddely what?" said Piglet.
"Pom," said Pooh. "I put that in to make it more hummy. The more it
goes, tiddely pom, the more——"
"Didn't you say snows?"
"Yes, but that was before."
"Before the tiddely pom?"
"It was a different tiddely pom," said Pooh, feeling rather muddled
now. "I'll sing it to you properly and then you'll see."
So he sang it again.

The more it
SNOWS-tiddely-pom,
The more it
GOES-tiddely-pom
The more it
GOES-tiddely-pom
On
Snowing.

And nobody
KNOWS-tiddely-pom,
How cold my
TOES-tiddely-pom
How cold my
TOES-tiddely-pom
Are
Growing.

He sang it like that, which is much the best way of singing it, and
when he had finished, he waited for Piglet to say that, of all the
Outdoor Hums for Snowy Weather he had ever heard, this was the
best. And, after thinking the matter out carefully, Piglet said:
"Pooh," he said solemnly, "it isn't the toes so much as the ears."
By this time they were getting near Eeyore's Gloomy Place, which
was where he lived, and as it was still very snowy behind Piglet's
ears, and he was getting tired of it, they turned into a little pine wood,
and sat down on the gate which led into it. They were out of the snow
now, but it was very cold, and to keep themselves warm they sang
Pooh's song right through six times, Piglet doing the tiddely-poms and
Pooh doing the rest of it, and both of them thumping on the top of the
gate with pieces of stick at the proper places. And in a little while they
felt much warmer, and were able to talk again.
"I've been thinking," said Pooh, "and what I've been thinking is this.
I've been thinking about Eeyore."
"What about Eeyore?"
"Well, poor Eeyore has nowhere to live."
"Nor he has," said Piglet.
"You have a house, Piglet, and I have a house, and they are very
good houses. And Christopher Robin has a house, and Owl and
Kanga and Rabbit have houses, and even Rabbit's friends and
relations have houses or somethings, but poor Eeyore has nothing.
So what I've been thinking is: Let's build him a house."
"That," said Piglet, "is a Grand Idea. Where shall we build it?"
"We build it here," said Pooh, "just by this wood, out of the wind,
because this is where I thought of it. And we will call this Pooh
Corner. And we will build an Eeyore House with sticks at Pooh Corner
for Eeyore."
"There was a heap of sticks on the other side of the wood," said
Piglet. "I saw them. Lots and lots. All piled up."
"Thank you, Piglet," said Pooh. "What you have just said will be a
Great Help to us, and because of it I could call this place
Poohanpiglet Corner if Pooh Corner didn't sound better, which it
does, being smaller and more like a corner. Come along."
So they got down off the gate and went round to the other side of the
wood to fetch the sticks.

Christopher Robin had spent the morning indoors going to Africa and
back, and he had just got off the boat and was wondering what it was
like outside, when who should come knocking at the door but Eeyore.
"Hallo, Eeyore," said Christopher Robin, as he opened the door and
came out. "How are you?"