Elementary Differential Equations and Boundary Value Problems

Boyce, William E.

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Elementary Differential Equations and Boundary Value  ...

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ISBN: 9781118323618, 830 blz., July 2017, Engels UITVERKOCHT!

Uitgever: Athenaeum Uitgeverij

beschrijving


  • A sound and accurate exposition of the elementary theory ofdifferential equations

  • Self-contained chapters allow instructors to customize theselection, order, and depth of chapters.

  • More than 450 problems are marked with a technology iconindicating those that are considered to be technologyintensive.

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inhoudsopgave

Chapter 1 Introduction 1


1.1 Some Basic Mathematical Models; Direction Fields 1


1.2 Solutions of Some Differential Equations 10


1.3 Classification of Differential Equations 19


1.4 Historical Remarks 26


Chapter 2 First Order Differential Equations 31


2.1 Linear Equations; Method of Integrating Factors 31


2.2 Separable Equations 42


2.3 Modeling with First Order Equations 51


2.4 Differences Between Linear and Nonlinear Equations 68


2.5 Autonomous Equations and Population Dynamics 78


2.6 Exact Equations and Integrating Factors 95


2.7 Numerical Approximations: Euler s Method 102


2.8 The Existence and Uniqueness Theorem 112


2.9 First Order Difference Equations 122


Chapter 3 Second Order Linear Equations 137


3.1 Homogeneous Equations with Constant Coefficients 137


3.2 Solutions of Linear Homogeneous Equations; the Wronskian145


3.3 Complex Roots of the Characteristic Equation 158


3.4 Repeated Roots; Reduction of Order 167


3.5 Nonhomogeneous Equations; Method of UndeterminedCoefficients 175


3.6 Variation of Parameters 186


3.7 Mechanical and Electrical Vibrations 192


3.8 Forced Vibrations 207


Chapter 4 Higher Order Linear Equations 221


4.1 General Theory of nth Order Linear Equations 221


4.2 Homogeneous Equations with Constant Coefficients 228


4.3 The Method of Undetermined Coefficients 236


4.4 The Method of Variation of Parameters 241


Chapter 5 Series Solutions of Second Order Linear Equations247


5.1 Review of Power Series 247


5.2 Series Solutions Near an Ordinary Point, Part I 254


5.3 Series Solutions Near an Ordinary Point, Part II 265


5.4 Euler Equations; Regular Singular Points 272


5.5 Series Solutions Near a Regular Singular Point, Part I282


5.6 Series Solutions Near a Regular Singular Point, Part II288


5.7 Bessel s Equation 296


Chapter 6 The Laplace Transform 309


6.1 Definition of the Laplace Transform 309


6.2 Solution of Initial Value Problems 317


6.3 Step Functions 327


6.4 Differential Equations with Discontinuous Forcing Functions336


6.5 Impulse Functions 343


6.6 The Convolution Integral 350


Chapter 7 Systems of First Order Linear Equations 359


7.1 Introduction 359


7.2 Review of Matrices 368


7.3 Systems of Linear Algebraic Equations; Linear Independence,Eigenvalues, Eigenvectors 378


7.4 Basic Theory of Systems of First Order Linear Equations390


7.5 Homogeneous Linear Systems with Constant Coefficients396


7.6 Complex Eigenvalues 408


7.7 Fundamental Matrices 421


7.8 Repeated Eigenvalues 429


7.9 Nonhomogeneous Linear Systems 440


Chapter 8 Numerical Methods 451


8.1 The Euler or Tangent Line Method 451


8.2 Improvements on the Euler Method 462


8.3 The Runge Kutta Method 468


8.4 Multistep Methods 472


8.5 Systems of First Order Equations 478


8.6 More on Errors; Stability 482


Chapter 9 Nonlinear Differential Equations and Stability495


9.1 The Phase Plane: Linear Systems 495


9.2 Autonomous Systems and Stability 508


9.3 Locally Linear Systems 519


9.4 Competing Species 531


9.5 Predator Prey Equations 544


9.6 Liapunov s Second Method 554


9.7 Periodic Solutions and Limit Cycles 565


9.8 Chaos and Strange Attractors: The Lorenz Equations 577


Chapter 10 Partial Differential Equations and Fourier Series589


10.1 Two-Point Boundary Value Problems 589


10.2 Fourier Series 596


10.3 The Fourier Convergence Theorem 607


10.4 Even and Odd Functions 614


10.5 Separation of Variables; Heat Conduction in a Rod 623


10.6 Other Heat Conduction Problems 632


10.7 TheWave Equation: Vibrations of an Elastic String 643


10.8 Laplace s Equation 658


AppendixA Derivation of the Heat Conduction Equation 669


Appendix B Derivation of theWave Equation 673


Chapter 11 Boundary Value Problems and Sturm LiouvilleTheory 677


11.1 The Occurrence of Two-Point Boundary Value Problems 677


11.2 Sturm Liouville Boundary Value Problems 685


11.3 Nonhomogeneous Boundary Value Problems 699


11.4 Singular Sturm Liouville Problems 714


11.5 Further Remarks on the Method of Separation of Variables: ABessel Series Expansion 721


11.6 Series of Orthogonal Functions: Mean Convergence 728


Answers to Problems 739


Index 799

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