Ordinary Differential Equations: Principles and Applications (Cambridge IISc Series)

Contents
Figures
Preface
Acknowledgement
1 Introduction and Examples: Physical Models
	1.1 A Brief General Introduction
	1.2 Physical and Other Models
		1.2.1 Population growth model
		1.2.2 An atomic waste disposal problem
		1.2.3 Mechanical vibration model
		1.2.4 Electrical circuit
		1.2.5 Satellite problem
		1.2.6 Flight trajectory problem
		1.2.7 Other examples
	1.3 Exercises
	1.4 Notes
2 Preliminaries
	2.1 Introduction
	2.2 Preliminaries from Real Analysis
		2.2.1 Convergence and uniform convergence
	2.3 Fixed Point Theorem
	2.4 Some Topics in Linear Algebra
		2.4.1 Euclidean space Rn
		2.4.2 Points versus vectors
		2.4.3 Linear operators
	2.5 Matrix Exponential eA and its Properties
		2.5.1 Diagonalizability and block diagonalizability
		2.5.2 Spectral analysis of A
		2.5.3 Computation of eJ for a Jordan block J
	2.6 Linear Dependence and Independence of Functions
	2.7 Exercises
	2.8 Notes
3 First and Second Order Linear Equations
	3.1 First Order Equations
		3.1.1 Initial and boundary value problems
		3.1.2 Concept of a solution
		3.1.3 First order linear equations
		3.1.4 Variable separable equations
	3.2 Exact Differential Equations
	3.3 Second Order Linear Equations
		3.3.1 Homogeneous SLDE (HSLDE)
		3.3.2 Linear equation with constant coefficients
		3.3.3 Non-homogeneous equation
		3.3.4 Green’s functions
	3.4 Partial Differential Equations and ODE
	3.5 Exercises
	3.6 Notes
4 General Theory of Initial Value Problems
	4.1 Introduction
		4.1.1 Well-posed problems
		4.1.2 Examples
	4.2 Sufficient Condition for Uniqueness of Solution
		4.2.1 A basic lemma
		4.2.2 Uniqueness theorem
	4.3 Sufficient Condition for Existence of Solution
		4.3.1 Cauchy–Peano existence theorem
		4.3.2 Existence and uniqueness by fixed point theorem
	4.4 Continuous Dependence of the Solution on Initial Data and Dynamics
	4.5 Continuation of a Solution into Larger Intervals and Maximal Interval of Existence
		4.5.1 Continuation of the solution outside the interval |t −t0| ≤ h
		4.5.2 Maximal interval of existence
	4.6 Existence and Uniqueness of a System of Equations
		4.6.1 Existence and uniqueness results for systems
	4.7 Exercises
	4.8 Notes
5 Linear Systems and Qualitative Analysis
	5.1 General nth Order Equations and Linear Systems
	5.2 Autonomous Homogeneous Systems
		5.2.1 Computation of etA in special cases
	5.3 Two-dimensional Systems
		5.3.1 Computation of eBj and etBj
	5.4 Stability Analysis
		5.4.1 Phase plane and phase portrait
		5.4.2 Dynamical system, flow, vector fields
		5.4.3 Equilibrium points and stability
	5.5 Higher Dimensional Systems
	5.6 Invariant Subspaces under the Flow etA
	5.7 Non-homogeneous, Autonomous Systems
		5.7.1 Solution to non-homogeneous systems (variation of parameters)
		5.7.2 Non-autonomous systems
	5.8 Exercises
	5.9 Notes
6 Series Solutions: Frobenius Theory
	6.1 Introduction
	6.2 Real Analytic Functions
	6.3 Equations with Analytic Coefficients
	6.4 Regular Singular Points
		6.4.1 Equations with regular singular points
	6.5 Exercises
	6.6 Notes
7 Regular Sturm–Liouville Theory
	7.1 Introduction
	7.2 Basic Result and Orthogonality
	7.3 Oscillation Results
		7.3.1 Comparison theorems
		7.3.2 Location of zeros
	7.4 Existence of Eigenfunctions
	7.5 Exercises
	7.6 Notes
8 Qualitative Theory
	8.1 Introduction
	8.2 General Definitions and Results
		8.2.1 Examples
	8.3 Liapunov Stability, Liapunov Function
		8.3.1 Linearization
		8.3.2 Examples
	8.4 Liapunov Function
	8.5 Invariant Subspaces and Manifolds
	8.6 Phase Plane Analysis
		8.6.1 Examples
	8.7 Periodic Orbits
	8.8 Exercises
	8.9 Notes
9 Two Point Boundary Value Problems
	9.1 Introduction
	9.2 Linear Problems
		9.2.1 BVP for linear systems
		9.2.2 Examples
	9.3 General Second Order Equations
		9.3.1 Examples
	9.4 Exercises
	9.5 Notes
10 First Order Partial Differential Equations: Method of Characteristics
	10.1 Linear Equations
	10.2 Quasi-linear Equations
	10.3 General First Order Equation in Two Variables
	10.4 Hamilton–Jacobi Equation
	10.5 Exercises
	10.6 Notes
Appendix A Poincar ` e–Bendixon and Leinard’s Theorems
	A.1 Introduction
	A.2 Poincar` e–Bendixon Theorems
		A.2.1 Intersection with transversals
	A.3 Leinard’s Theorem
Bibliography
Index