1.1 Dimensional Analysis, l

1.2 Expansions, 10

1.3 Gauge Functions, 12

1.4 Order Symbols, 17

1.5 Asymptotic Series, 18

1.6 Asymptotic Expansions and Sequences, 22

1.7 Convergent Versus Asymptotic Series, 23

1.8 Elementary Operations on Asymptotic Expansions, 24

2 ALGEBRAIC EQUATIONS 28

2.1 Quadratic Equations, 28

2.2 Cubic Equations, 39

2.3 Higher-Order Equations, 43

2.4 Transcendental Equations, 45

3 INTEGRALS 51

3.1 Expansion of Integrands, 52

3.2 Integration by Parts, 56

3.3 Laplace's Method, 65

3.4 The Method of Stationary Phase, 79

3.5 The Method of Steepest Descent, 88

4 THE DUFFING EQUATION 107

4.1 The Straightforward Expansion, 109

4.2 Exact Solution, 113

4.3 The Lindstedt-Poincaré Technique, 118

4.4 The Method of Renormalization, 121

4.5 The Method of Multiple Scales, 122

4.6 Variation of Parameters, 127

4.7 The Method of Averaging, 129

5 THE LINEAR DAMPED OSCILLATOR 134

5.1 The Straightforward Expansion, 135

5.2 Exact Solution, 136

5.3 The Lindstedt-Poincaré Technique, 139

5.4 The Method of Multiple Scales, 142

5.5 The Method of Averaging, 144

6 SELF-EXCITED OSCILLATORS 147

6.1 The Straightforward Expansion, 148

6.2 The Method of Renormalization, 151

6.3 The Method of Multiple Scales, 152

6.4 The Method of Averaging, 155

7 SYSTEMS WITH QUADRATIC AND CUBIC NONLINEARITIES 159

7.1 The Straightforward Expansion, 160

7.2 The Method of Renormalization, 162

7.3 The Lindstedt-Poincaré Technique, 164

7.4 The Method of Multiple Scales, 166

7.5 The Method of Averaging, 168

7.6 The Generalized Method of Averaging, 169

7.7 The Krylov-Bogoliubov-Mitropolsky Technique, 173

8 GENERAL WEAKLY NONLINEAR SYSTEMS 177

8.1 The Straightforward Expansion, 177

8.2 The Method of Renormalization, 179

8.3 The Method of Multiple Scales, 181

8.4 The Method of Averaging, 182

8.5 Applications, 184

9 FORCED OSCILLATIONS OF THE DUFFING EQUATION 190

9.1 The Straightforward Expansion, 191

9.2 The Method of Multiple Scales, 193

9.2.1 Secondary Resonances, 193

9.2.2 Primary Resonance, 205

9.3 The Method of Averaging, 209

9.3.1 Secondary Resonances, 209

9.3.2 Primary Resonance, 212

10 MULTIFREQUENCY EXCITATIONS 216

10.1 The Straightforward Expansion, 216

10.2 The Method of Multiple Scales, 219

10.3 The Method of Averaging, 226

11 THE MATHIEU EQUATION 234

11.1 The Straightforward Expansion, 235

11.2 The Floquet Theory, 236

11.3 The Method of Strained Parameters, 243

11.4 Whittaker's Method, 247

11.5 The Method of Multiple Scales, 249

11.6 The Method of Averaging, 253

12 BOUNDARY-LAYER PROBLEMS 257

12.1 A Simple Example, 257

12.2 The Method of Multiple Scales, 268

12.3 The Method of Matched Asymptotic Expansions, 270

12.4 Higher Approximations, 279

12.5 Equations with Variable Coefficients, 284

12.6 Problems with Two Boundary Layers, 296

12.7 Multiple Decks, 304

12.8 Nonlinear Problems, 307

13 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS 325

13.1 First-Order Scalar Equations, 326

13.2 Second-Order Equations, 329

13.3 Solutions Near Regular Singular Points, 331

13.4 Singularity at Infinity, 342

13.5 Solutions Near an Irregular Singular Point, 344

14 DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER 360

14.1 The WKB Approximation, 361

14.2 The Liouville-Green Transformation, 364

14.3 Eigenvalue Problems, 366

14.4 Equations with Slowly Varying Coefficients, 369

14.5 Turning-Point Problems, 370

14.6 The Langer Transformation, 375

14.7 Eigenvalue Problems with Turning Points, 379

15 SOLVABILITY CONDITIONS 388

15.1 Algebraic Equations, 389

15.2 Nonlinear Vibrations of Two-Degree-of-Freedom Gyroscopic Systems, 394

15.3 Parametrically Excited Gyroscopic Systems, 397

15.4 Second-Order Differential Systems, 401

15.5 General Boundary Conditions, 406

15.6 A Simple Eigenvalue Problem, 412

15.7 A Degenerate Eigenvalue Problem, 414

15.8 Acoustic Waves in a Duct with Sinusoidal Walls, 418

15.9 Vibrations of Nearly Circular Membranes, 426

15.10 A Fourth-Order Differential System, 432

15.11 General Fourth-Order Differential Systems, 438

15.12 A Fourth-Order Eigenvalue Problem, 441

15.13 A Differential System of Equations, 445

15.14 General Differential Systems of First-Order Equations, 447

15.15 Differential Systems with Interfacial Boundary Conditions, 452

15.16 Integral Equations, 454

15.17 Partial-Differential Equations, 458

APPENDIX A TRIGONOMETRIC IDENTITIES 472

APPENDIX B LINEAR ORDINARY-DIFFERENTIAL EQUATIONS 480

BIBLIOGRAPHY 501

INDEX 507