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CHAPTER 1．EXISTENCE AND UNIQUENESS OF SOLUTIONS1

1．Existence of Solutions1

2．Uniqueness of Solutions8

3．The Method of Successive Approximations11

4．Continuation of Solutions13

5．Systems of Differential Equations15

6．The nth-order Equation21

7．Dependence of Solutions on Initial Conditions and Parameters22

8．Complex Systems32

Problems37

CHAPTER 2．EXISTENCE AND UNIQUENESS OF SOLUTIONS(continued)42

1．Extension of the Idea of a Solution,Maximum and Minimum Solutions42

2．Further Uniqueness Results48

3．Uniqueness and Successive Approximations53

4．Variation of Solutions with Respect to Initial Conditions and Parameters57

Problems60

CHAPTER 3．LINEAR DIFFERENTIAL EQUATIONS62

1．Preliminary Definitions and Notations62

2．Linear Homogeneous Systems67

3．Nonhomogeneous Linear Systems74

4．Linear Systems with Constant Coefficients75

5．Linear Systems with Periodic Coefficients78

6．Linear Differential Equstions of Order n81

7．Linear Equations with Analytic Coefficients90

8．Asymptotic Behavior of the Solutions of Certain Linear Systems91

Problems97

CHAPTER 4．LINEAR SYSTEMS WITH ISOLATED SINGULARITIES:SINGULARITIES OF THE FIRST KIND108

1．Introduction108

2．Classification of Singularities111

3．Formal Solutions114

4．Structure of Fundamental Matrices118

5．The Equation of the nth Order122

6．Singularities at Infinity127

7．An Example:the Second-order Equation130

8．The Frobenius Method132

Problems135

CHAPTER 5．LINEAR SYSTEMS WITH ISOLATED SINGULARITIES:SINGULARITIES OF THE SECOND KIND138

1．Introduction138

2．Formal Solutions141

3．Asymptotic Series148

4．Existence of Solutions Which Have the Formal Solutions as Asymptotic Expansions—the Real Case151

5．The Asymptotic Nature of the Formal Solutions in the Complex Case161

6．The Case Where A0 Has Multiple Characteristic Roots167

7．Irregular Singular Points of an nth-order Equation169

8．The Laplace Integral and Asymptotic Series170

Problems173

CHAPTER 6．ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS CONTAINING A LARGE PARAMETER174

1．Introduction174

2．Formal Solutions175

3．Asymptotic Behavior of Solutions178

4．The Case of Equal Characteristic Roots182

5．The nth-order Equation182

Problems184

CHAPTER 7．SELF-ADJOINT EIGENVALUE PROBLEMS ON A FINITE INTERVAL186

1．Introduction186

2．Self-adjoint Eigenvalue Problems188

3．The Existence of Eigenvalues193

4．The Expansion and Completeness Theorems197

Problems201

CHAPTER 8．OSCILLATION AND COMPARISON THEOREMS FOR SECOND-ORDER LINEAR EQUATIONS AND APPLICATIONS208

1．Comparison Theorems208

2．Existence of Eigenvalues211

3．Periodic Boundary Conditions213

4．Stability Regions of Second-order Equations with Periodic Coefficients218

Problems220

CHAPTER 9．SINGULAR SELF-ADJOINT BOUNDARY-VALUE PROBLEMS FOR SECOND-ORDER EQUATIONS222

1．Introduction222

2．The Limit-point and Limit-circle Cases225

3．The Completeness and Expansion Theorems in the Limit-point Case at Infinity231

4．The Limit-circle Case at Infinity242

5．Singular Behavior at Both Ends of an Interval246

Problems254

CHAPTER 10．SINGULAR SELF-ADJOINT BOUNDARY-VALUE PROBLEMS FOR nTH-ORDER EQUATIONS261

1．Introduction261

2．The Expansion Theorem and Parseval Equality262

3．The Inverse-transform Theorem and the Uniqueness of the Spectral Matrix265

4．Green＇s Function272

5．Representation of the Spectral Matrix by Green＇s Function278

Problems281

CHAPTER 11．ALGEBRAIC PROPERTIES OF LINEAR BOUNDARY-VALUE PROBLEMS ON A FINITE INTERVAL284

1．Introduction284

2．The Boundary-form Formula286

3．Homogeneous Boundary-value Problems and Adjoint Problems288

4．Nonhomogeneous Boundary-value Problems and Green＇s Function294

Problems297

CHAPTER 12．NON-SELF-ADJOINT BOUNDARY-VALUE PROBLEMS298

1．Introduction298

2．Green＇s Function and the Expansion Theorem for the Case Lx=-x″300

3．Green＇s Function and the Expansion Theorem for the Case Lx=-x″+q(t)x305

4．The nth-order Case308

5．The Form of the Expansion310

Problems312

CHAPTER 13．ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS:STABILITY314

1．Asymptotic Stability314

2．First Variation:Orbital Stability321

3．Asymptotic Behavior of a System327

4．Conditional Stability329

5．Behavior of Solutions off the Stable Manifold340

Problems344

CHAPTER 14．PERTURBATION OF SYSTEMS HAVING A PERIODIC SOLUTION348

1．Nonautonomous Systems348

2．Autonomous Systems352

3．Perturbation of a Linear System with a Periodic Solution in the Nonautonomous Case356

4．Perturbation of an Autonomous System with a Vanishing Jacobian364

Problems370

CHAPTER 15．PERTURBATION THEORY OF TWO-DIMENSIONAL REAL AUTONOMOUS SYSTEMS371

1．Two-dimensional Linear Systems371

2．Perturbations of Two-dimensional Linear Systems375

3．Proper Nodes and Proper Spiral Points377

4．Centers381

5．Improper Nodes384

6．Saddle Points387

Problems388

CHAPTER 16 THE POINCARE-BENDIXSON THEORY OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS389

1．Limit Sets of an Orbit389

2．The Poincaré-Bendixson Theorem391

3．Limit Sets with Critical Points394

4．The Index of an Isolated Critical Point398

5．The Index of Simple Critical Point400

Problems402

CHAPTER 17．DIFFERENTIAL EQUATIONS ON A TORUS404

1．Introduction404

2．The Rotation Number405

3．The Cluster Set408

4．The Ergodic Case409

5．Characterization of Solutions in the Ergodic Case413

6．A System of Two Equations415

REFERENCES417

INDEX423