viii CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace's Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469viii ● CONTENTS 10 PLANE AUTONOMOUS SYSTEMS 363 10.1 Autonomous Systems 364 10.2 Stability of Linear Systems 370 10.3 Linearization and Local Stability 378 10.4 Autonomous Systems as Mathematical Models 388 CHAPTER 10 IN REVIEW 395 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 397 11.1 Orthogonal Functions 398 11.2 Fourier Series 403 11.3 Fourier Cosine and Sine Series 408 11.4 Sturm-Liouville Problem 416 11.5 Bessel and Legendre Series 423 11.5.1 Fourier-Bessel Series 424 11.5.2 Fourier-Legendre Series 427 CHAPTER 11 IN REVIEW 430 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 432 12.1 Separable Partial Differential Equations 433 12.2 Classical PDEs and Boundary-Value Problems 437 12.3 Heat Equation 443 12.4 Wave Equation 445 12.5 Laplace’s Equation 450 12.6 Nonhomogeneous Boundary-Value Problems 455 12.7 Orthogonal Series Expansions 461 12.8 Higher-Dimensional Problems 466 CHAPTER 12 IN REVIEW 469