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.
Author(s): A. K. Nandakumaran, P. S. Datti, Raju K. George
Series: Cambridge-IISc Series
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 349
Contents......Page 8
Figures......Page 14
Preface......Page 15
Acknowledgement......Page 19
1.1 A Brief General Introduction......Page 22
1.2.1 Population growth model......Page 24
1.2.2 An atomic waste disposal problem......Page 27
1.2.3 Mechanical vibration model......Page 29
1.2.4 Electrical circuit......Page 30
1.2.5 Satellite problem......Page 31
1.2.6 Flight trajectory problem......Page 34
1.2.7 Other examples......Page 35
1.3 Exercises......Page 40
1.4 Notes......Page 42
2.2.1 Convergence and uniform convergence......Page 43
2.3 Fixed Point Theorem......Page 55
2.4 Some Topics in Linear Algebra......Page 59
2.4.1 Euclidean space Rn......Page 62
2.4.3 Linear operators......Page 63
2.5 Matrix Exponential eA and its Properties......Page 64
2.5.1 Diagonalizability and block diagonalizability......Page 67
2.5.2 Spectral analysis of A......Page 69
2.5.3 Computation of eJ for a Jordan block J......Page 72
2.6 Linear Dependence and Independence of Functions......Page 74
2.7 Exercises......Page 75
2.8 Notes......Page 76
3.1 First Order Equations......Page 77
3.1.1 Initial and boundary value problems......Page 78
3.1.2 Concept of a solution......Page 80
3.1.3 First order linear equations......Page 81
3.1.4 Variable separable equations......Page 86
3.2 Exact Differential Equations......Page 87
3.3 Second Order Linear Equations......Page 93
3.3.1 Homogeneous SLDE (HSLDE)......Page 95
3.3.2 Linear equation with constant coefficients......Page 98
3.3.3 Non-homogeneous equation......Page 100
3.3.4 Green’s functions......Page 109
3.4 Partial Differential Equations and ODE......Page 110
3.5 Exercises......Page 114
3.6 Notes......Page 118
4.1.1 Well-posed problems......Page 120
4.1.2 Examples......Page 121
4.2.1 A basic lemma......Page 124
4.2.2 Uniqueness theorem......Page 127
4.3 Sufficient Condition for Existence of Solution......Page 128
4.3.1 Cauchy–Peano existence theorem......Page 133
4.3.2 Existence and uniqueness by fixed point theorem......Page 137
4.4 Continuous Dependence of the Solution on Initial Data and Dynamics......Page 140
4.5 Continuation of a Solution into Larger Intervals and Maximal Interval of Existence......Page 141
4.5.1 Continuation of the solution outside the interval |t −t0| ≤ h......Page 142
4.5.2 Maximal interval of existence......Page 144
4.6 Existence and Uniqueness of a System of Equations......Page 146
4.6.1 Existence and uniqueness results for systems......Page 148
4.7 Exercises......Page 151
4.8 Notes......Page 154
5.1 General nth Order Equations and Linear Systems......Page 155
5.2 Autonomous Homogeneous Systems......Page 157
5.2.1 Computation of etA in special cases......Page 158
5.3 Two-dimensional Systems......Page 160
5.3.1 Computation of eBj and etBj......Page 161
5.4.1 Phase plane and phase portrait......Page 164
5.4.2 Dynamical system, flow, vector fields......Page 166
5.4.3 Equilibrium points and stability......Page 168
5.5 Higher Dimensional Systems......Page 176
5.6 Invariant Subspaces under the Flow etA......Page 186
5.7 Non-homogeneous, Autonomous Systems......Page 188
5.7.1 Solution to non-homogeneous systems (variation of parameters)......Page 189
5.7.2 Non-autonomous systems......Page 190
5.8 Exercises......Page 196
5.9 Notes......Page 200
6.2 Real Analytic Functions......Page 201
6.3 Equations with Analytic Coefficients......Page 204
6.4 Regular Singular Points......Page 210
6.4.1 Equations with regular singular points......Page 211
6.5 Exercises......Page 219
6.6 Notes......Page 221
7.1 Introduction......Page 222
7.2 Basic Result and Orthogonality......Page 224
7.3 Oscillation Results......Page 229
7.3.1 Comparison theorems......Page 232
7.3.2 Location of zeros......Page 239
7.4 Existence of Eigenfunctions......Page 240
7.5 Exercises......Page 241
7.6 Notes......Page 243
8.1 Introduction......Page 244
8.2 General Definitions and Results......Page 245
8.2.1 Examples......Page 249
8.3 Liapunov Stability, Liapunov Function......Page 250
8.3.1 Linearization......Page 251
8.3.2 Examples......Page 254
8.4 Liapunov Function......Page 259
8.5 Invariant Subspaces and Manifolds......Page 266
8.6.1 Examples......Page 269
8.7 Periodic Orbits......Page 274
8.8 Exercises......Page 285
8.9 Notes......Page 287
9.1 Introduction......Page 288
9.2 Linear Problems......Page 290
9.2.1 BVP for linear systems......Page 295
9.2.2 Examples......Page 296
9.3 General Second Order Equations......Page 297
9.3.1 Examples......Page 301
9.5 Notes......Page 305
10.1 Linear Equations......Page 307
10.2 Quasi-linear Equations......Page 311
10.3 General First Order Equation in Two Variables......Page 316
10.4 Hamilton–Jacobi Equation......Page 323
10.5 Exercises......Page 325
10.6 Notes......Page 327
A.1 Introduction......Page 328
A.2.1 Intersection with transversals......Page 331
A.3 Leinard’s Theorem......Page 337
Bibliography......Page 342
Index......Page 346
