Differential Equations: A Linear Algebra Approach

626 Pages·2021·5.247 MB·English

by  Anindya Dey

Differential Equations
A Linear Algebra Approach
Anindya Dey
Levant Books
India
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Dedicated to
My wife Aparna & daughters Riddhi, Anandi

Preface
‘There is nothing so practical as a good theory’
−−Kurt Lewin
Mathematics plays a pivotal role in many scientific and engineering
disciplines and ordinary differential equation is certainly an important
branch of it. When you publish either a text book or a reference book
on such a classical topic, you have to face the obvious question : ‘why
yet another?’
The idea of preparing a book on this classical topic came to my
mind in the process of teaching this subject at the undergraduate level
for more than a decade. As experienced, our undergraduate students
are by and large accustomed to mechanical workout of gritty problems
but they fail to understand the legitimacy of the methods in vogue and
more surprisingly cannot appreciate the geometrical arguments or phys-
ical applications of this powerful tool. However, should not the learners
be solely blamed for this setback as dearth of a book presenting the
rigorous theoretical aspects concisely and parallelly citing a plenty of
practical applications of the differential equations is also a prime reason
for that shortcoming. The existing texts that I had the opportunity to
have a glance of, either concentrate on a serious note of pedantic dis-
cussion of the deep-rooted results emerging from mathematical analysis
(e,g existence, uniqueness, boundedness, stability of solutions) or are
very informal in the theoretical aspects while elaborative on catering
various routine techniques of attacking the problems. My decade long
teaching experience prompts that most of the popular books on this
subject neither point out the origin of a particular method nor its scope
and limitations. The symmetry aspect of an ode, the interplay of linear
algebra and linear differential equations, the connection between an ex-
act ode and a conservative vector field are not usually addressed. This
leads to a delinked study of this powerful subject and an ardent reader
does not get the much desired interest in the topic.
Theitemsencompassedinthisbookarecommontoanyintroductory
course delivered at the graduate level and partly at the postgraduate
level, but it has been dished out in a new mould. This volume is a
self-contained introduction to the field of ode with an emphasis on the
underlying role played by linear algebra in delving into the solutions of
linearodes. Thereaderonlyneedsapreliminaryknowledgeofcalculusin
Randelementary linearalgebratopursuethebook. PersonallyIbelieve
that the success of a book lies not in how much topic it encompassed
in a pedantic style, rather how much interest could it kindle in the
learners through its presentation. This motivated me to discuss the
theories and its applications side-by-side. Often one will find prolonged
remarks following a particular theory or even an illustrative example
since through suitable examples the scope and limitations of a method
comes to surface. However, as already told, it was not possible for me
to confine all the study material within the grasp of the undergraduate
students for whom the venture was initially aimed [chapter 4, parts of
chapter 6andchapter 8andchapter 9arebeyondtraditionalgraduation
courses of Indian Universities]. Even with all these drawbacks if the
present work could serve the learners at least to some extent, I shall
feel rewarded. I look ahead for more constructive suggestions from the
students as well as teachers so that I may patch up the lacunae or
incompleteness and streamline the ideas better and user-friendly.
It’s a pleasure to acknowledge the help I received in bringing this
projectto fruition. First of allI mustthankall mystudentswhoatleast
helped me learn the subject a bit and instigated me to go beyond the
barriers of protocol teaching despite many constraints. I also thank my
friendsandcolleagues whoprocuredalotofsuggestions andcorrections.
Should I express thanks to Levant Books, specially Dr. Milinda De, Mr.
Debasis Auddy and their team for taking the brunt of shaping up the
bookthroughalongstintofnearlythreeyears. LastbutnotleastImust
express my deepest thank to my wife and children whose forebearence
over the years of ups and downs has been habitual.
Kolkata AnindyaDey
April 2017 Assistant Professor,
Dept. of Mathematics
St. Xavier’s College Kolkata
Contents
Preface v
1 A Prelude to Differential Equations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Formulation of Differential Equation−Its Significance. . . 7
1.3 Classification of Solutions : General , Particular and Sin-
gular Solutions . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 More about Solutions of an ODE . . . . . . . . . . . . . . 20
1.5 Existence-Uniqueness Theorem for Cauchy Problem. . . . 24
1.6 Importance of Lipschitz’s condition involved in existence
uniqueness theorem in the light of a comparative study
between Radiactive decay and Leaky bucket problems : . 30
1.7 First Order Ode and some of its Qualitative Aspects . . . 34
2 Equations of First Order and First Degree 41
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Exact Differential Equation . . . . . . . . . . . . . . . . . 43
2.3 Homogeneous Differential Equations . . . . . . . . . . . . 61
2.4 Integrating Factor . . . . . . . . . . . . . . . . . . . . . . 70
2.5 Linear Equations and Bernoulli Equations . . . . . . . . . 80
2.6 Integrating Factors Revisited . . . . . . . . . . . . . . . . 93
2.7 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . 95
2.8 Application of Differential Equations of First Order . . . 101
2.9 Orthogonal Trajectories and Oblique Trajectories . . . . . 113
3 A Class of First Order Non-Linear Odes 130
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2 Non-linear First Order Ode Solvable for p . . . . . . . . . 130
3.3 Non-linear Ode Solvable for y . . . . . . . . . . . . . . . 134
3.4 Non-linear Ode Solvable for x . . . . . . . . . . . . . . . . 135
3.5 Existence and Uniqueness Problem . . . . . . . . . . . . 138
3.6 Envelopes and Other Loci . . . . . . . . . . . . . . . . . 144
3.7 Clairaut’s Equation and Lagrange’s Equation . . . . . . . 154
4 Linear Algebraic Framework
in Differential Equations 176
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.2 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.3 Linear Maps or Transformations . . . . . . . . . . . . . . 184
4.4 Normed Linear Space . . . . . . . . . . . . . . . . . . . . 185
4.5 Bounded Linear Transformation . . . . . . . . . . . . . . . 191
4.6 Invertible Operators . . . . . . . . . . . . . . . . . . . . . 197
5 Differential Equations of Higher Order 209
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.2 Theoretical Aspects . . . . . . . . . . . . . . . . . . . . . 209
5.3 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.4 Working Rules for Homogeneous Linear Ode. . . . . . . . 225
5.5 Few Theoretical Results from Linear Algebra. . . . . . . . 231
1
5.6 Symbolic Operator and Particular Integral . . . . . . 233
L(D)
5.7 Method of Variation of Parameters . . . . . . . . . . . . . 238
5.8 Special Methods for finding Particular Integrals . . . . . . 251
5.9 Method of Undetermined Co-efficients . . . . . . . . . . . 262
5.10 Fourier Series Method for Particular Integrals . . . . . . . 274
6 Second Order Linear Ode: Solution Techniques & Qual-
itative Analysis 284
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 284
6.2 Reduction of Order Method (D’Alembert’s Method) . . . 285
6.3 Method of Inspection for finding one Integral . . . . . . . 291
6.4 Transformation of Second Order Ode by changing the
Independent Variable . . . . . . . . . . . . . . . . . . . . 294
6.5 Transformation of a Second order Ode by changing the
Dependent Variable: . . . . . . . . . . . . . . . . . . . . . 306
6.6 Qualitative AspectsofSecondOrderDifferential Equations.309
6.7 Exact Second Order Differential Equations. . . . . . . . . 318
6.8 Adjoint Equation and Self-adjoint Odes . . . . . . . . . . 323
6.9 Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . 332
6.10 Green’s Function Approach to IVP . . . . . . . . . . . . . 357
6.11 Green’s Function Approach to BVP . . . . . . . . . . . . 362
7 Laplace Transformations in Ordinary Differential Equa-
tions 380
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 380
7.2 Definition and Anatomy of Laplace Transform. . . . . . . 380
7.3 LaplacetransformationtechniqueofsolvingOrdinaryDif-
ferential Equations . . . . . . . . . . . . . . . . . . . . . . 408
8 Series Solutions of Linear Differential Equations 438
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 438
8.2 Review of Power Series . . . . . . . . . . . . . . . . . . . . 439
8.3 Solutions about Ordinary Points in the Domain . . . . . . 444
8.4 Solution about Regular Singular Points . . . . . . . . . . 461
8.5 Frobenius Method . . . . . . . . . . . . . . . . . . . . . . 468
8.6 Hypergeometric Equation . . . . . . . . . . . . . . . . . . 509
8.7 Irregular singular points . . . . . . . . . . . . . . . . . . . 514
9 Solving Linear Systems by Matrix Methods 528
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 528
9.2 Eigenvalue Problems of a square matrix : Diagonalisability534