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CHAPTER 1．EXISTENCE AND UNIQUENESS OF SOLUTIONS 1

1．Existence of Solutions 1

2．Uniqueness of Solutions 8

3．The Method of Successive Approximations 11

4．Continuation of Solutions 13

5．Systems of Differential Equations 15

6．The nth-order Equation 21

7．Dependence of Solutions on Initial Conditions and Parameters 22

8．Complex Systems 32

Problems 37

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

1．Extension of the Idea of a Solution,Maximum and Minimum Solutions 42

2．Further Uniqueness Results 48

3．Uniqueness and Successive Approximations 53

4．Variation of Solutions with Respect to Initial Conditions and Parameters 57

Problems 60

CHAPTER 3．LINEAR DIFFERENTIAL EQUATIONS 62

1．Preliminary Definitions and Notations 62

2．Linear Homogeneous Systems 67

3．Nonhomogeneous Linear Systems 74

4．Linear Systems with Constant Coefficients 75

5．Linear Systems with Periodic Coefficients 78

6．Linear Differential Equstions of Order n 81

7．Linear Equations with Analytic Coefficients 90

8．Asymptotic Behavior of the Solutions of Certain Linear Systems 91

Problems 97

CHAPTER 4．LINEAR SYSTEMS WITH ISOLATED SINGULARITIES:SINGULARITIES OF THE FIRST KIND 108

1．Introduction 108

2．Classification of Singularities 111

3．Formal Solutions 114

4．Structure of Fundamental Matrices 118

5．The Equation of the nth Order 122

6．Singularities at Infinity 127

7．An Example:the Second-order Equation 130

8．The Frobenius Method 132

Problems 135

CHAPTER 5．LINEAR SYSTEMS WITH ISOLATED SINGULARITIES:SINGULARITIES OF THE SECOND KIND 138

1．Introduction 138

2．Formal Solutions 141

3．Asymptotic Series 148

4．Existence of Solutions Which Have the Formal Solutions as Asymptotic Expansions—the Real Case 151

5．The Asymptotic Nature of the Formal Solutions in the Complex Case 161

6．The Case Where A0 Has Multiple Characteristic Roots 167

7．Irregular Singular Points of an nth-order Equation 169

8．The Laplace Integral and Asymptotic Series 170

Problems 173

CHAPTER 6．ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS CONTAINING A LARGE PARAMETER 174

1．Introduction 174

2．Formal Solutions 175

3．Asymptotic Behavior of Solutions 178

4．The Case of Equal Characteristic Roots 182

5．The nth-order Equation 182

Problems 184

CHAPTER 7．SELF-ADJOINT EIGENVALUE PROBLEMS ON A FINITE INTERVAL 186

1．Introduction 186

2．Self-adjoint Eigenvalue Problems 188

3．The Existence of Eigenvalues 193

4．The Expansion and Completeness Theorems 197

Problems 201

CHAPTER 8．OSCILLATION AND COMPARISON THEOREMS FOR SECOND-ORDER LINEAR EQUATIONS AND APPLICATIONS 208

1．Comparison Theorems 208

2．Existence of Eigenvalues 211

3．Periodic Boundary Conditions 213

4．Stability Regions of Second-order Equations with Periodic Coefficients 218

Problems 220

CHAPTER 9．SINGULAR SELF-ADJOINT BOUNDARY-VALUE PROBLEMS FOR SECOND-ORDER EQUATIONS 222

1．Introduction 222

2．The Limit-point and Limit-circle Cases 225

3．The Completeness and Expansion Theorems in the Limit-point Case at Infinity 231

4．The Limit-circle Case at Infinity 242

5．Singular Behavior at Both Ends of an Interval 246

Problems 254

CHAPTER 10．SINGULAR SELF-ADJOINT BOUNDARY-VALUE PROBLEMS FOR nTH-ORDER EQUATIONS 261

1．Introduction 261

2．The Expansion Theorem and Parseval Equality 262

3．The Inverse-transform Theorem and the Uniqueness of the Spectral Matrix 265

4．Green＇s Function 272

5．Representation of the Spectral Matrix by Green＇s Function 278

Problems 281

CHAPTER 11．ALGEBRAIC PROPERTIES OF LINEAR BOUNDARY-VALUE PROBLEMS ON A FINITE INTERVAL 284

1．Introduction 284

2．The Boundary-form Formula 286

3．Homogeneous Boundary-value Problems and Adjoint Problems 288

4．Nonhomogeneous Boundary-value Problems and Green＇s Function 294

Problems 297

CHAPTER 12．NON-SELF-ADJOINT BOUNDARY-VALUE PROBLEMS 298

1．Introduction 298

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)x 305

4．The nth-order Case 308

5．The Form of the Expansion 310

Problems 312

CHAPTER 13．ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS:STABILITY 314

1．Asymptotic Stability 314

2．First Variation:Orbital Stability 321

3．Asymptotic Behavior of a System 327

4．Conditional Stability 329

5．Behavior of Solutions off the Stable Manifold 340

Problems 344

CHAPTER 14．PERTURBATION OF SYSTEMS HAVING A PERIODIC SOLUTION 348

1．Nonautonomous Systems 348

2．Autonomous Systems 352

3．Perturbation of a Linear System with a Periodic Solution in the Nonautonomous Case 356

4．Perturbation of an Autonomous System with a Vanishing Jacobian 364

Problems 370

CHAPTER 15．PERTURBATION THEORY OF TWO-DIMENSIONAL REAL AUTONOMOUS SYSTEMS 371

1．Two-dimensional Linear Systems 371

2．Perturbations of Two-dimensional Linear Systems 375

3．Proper Nodes and Proper Spiral Points 377

4．Centers 381

5．Improper Nodes 384

6．Saddle Points 387

Problems 388

CHAPTER 16 THE POINCARE-BENDIXSON THEORY OF TWO-DIMENSIONAL AUTONOMOUS SYSTEMS 389

1．Limit Sets of an Orbit 389

2．The Poincaré-Bendixson Theorem 391

3．Limit Sets with Critical Points 394

4．The Index of an Isolated Critical Point 398

5．The Index of Simple Critical Point 400

Problems 402

CHAPTER 17．DIFFERENTIAL EQUATIONS ON A TORUS 404

1．Introduction 404

2．The Rotation Number 405

3．The Cluster Set 408

4．The Ergodic Case 409

5．Characterization of Solutions in the Ergodic Case 413

6．A System of Two Equations 415

REFERENCES 417

INDEX 423