1.1 Basic Concepts. Modeling 2

1.2 Geometric Meaning of y ƒ(x, y). Direction Fields, Euler’s Method 9

1.3 Separable ODEs. Modeling 12

1.4 Exact ODEs. Integrating Factors 20

1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27

1.6 Orthogonal Trajectories. Optional 36

1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38

2.1 Homogeneous Linear ODEs of Second Order 46

2.2 Homogeneous Linear ODEs with Constant Coefficients 53

2.3 Differential Operators. Optional 60

2.4 Modeling of Free Oscillations of a Mass–Spring System 62

2.5 Euler–Cauchy Equations 71

2.6 Existence and Uniqueness of Solutions. Wronskian 74

2.7 Nonhomogeneous ODEs 79

2.8 Modeling: Forced Oscillations. Resonance 85

2.9 Modeling: Electric Circuits 93

2.10 Solution by Variation of Parameters 99

3.1 Homogeneous Linear ODEs 105

3.2 Homogeneous Linear ODEs with Constant Coefficients 111

3.3 Nonhomogeneous Linear ODEs 116

4.0 For Reference: Basics of Matrices and Vectors 124

4.1 Systems of ODEs as Models in Engineering Applications 130

4.2 Basic Theory of Systems of ODEs. Wronskian 137

4.3 Constant-Coefficient Systems. Phase Plane Method 140

4.4 Criteria for Critical Points. Stability 148

4.5 Qualitative Methods for Nonlinear Systems 152

4.6 Nonhomogeneous Linear Systems of ODEs 160

5.1 Power Series Method 167

5.2 Legendre's Equation. Legendre Polynomials Pn(x) 175

5.3 Extended Power Series Method: Frobenius Method 180

5.4 Bessel’s Equation. Bessel Functions (x) 187

5.5 Bessel Functions of the Y (x). General Solution 196

6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) 204

6.2 Transforms of Derivatives and Integrals. ODEs 211

6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) 217

6.4 Short Impulses. Dirac's Delta Function. Partial Fractions 225

6.5 Convolution. Integral Equations 232

6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients 238

6.7 Systems of ODEs 242

6.8 Laplace Transform: General Formulas 248

6.9 Table of Laplace Transforms 249

7.1 Matrices, Vectors: Addition and Scalar Multiplication 257

7.2 Matrix Multiplication 263

7.3 Linear Systems of Equations. Gauss Elimination 272

7.4 Linear Independence. Rank of a Matrix. Vector Space 282

7.5 Solutions of Linear Systems: Existence, Uniqueness 288

7.6 For Reference: Second- and Third-Order Determinants 291

7.7 Determinants. Cramer’s Rule 293

7.8 Inverse of a Matrix. Gauss–Jordan Elimination 301

7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309

8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors 323

8.2 Some Applications of Eigenvalue Problems 329

8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334

8.4 Eigenbases. Diagonalization. Quadratic Forms 339

8.5 Complex Matrices and Forms. Optional 346

9.1 Vectors in 2-Space and 3-Space 354

9.2 Inner Product (Dot Product) 361

9.3 Vector Product (Cross Product) 368

9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375

9.5 Curves. Arc Length. Curvature. Torsion 381

9.6 Calculus Review: Functions of Several Variables. Optional 392

9.7 Gradient of a Scalar Field. Directional Derivative 395

9.8 Divergence of a Vector Field 403

9.9 Curl of a Vector Field 406

10.1 Line Integrals 413

10.2 Path Independence of Line Integrals 419

10.3 Calculus Review: Double Integrals. Optional 426

10.4 Green’s Theorem in the Plane 433

10.5 Surfaces for Surface Integrals 439

10.6 Surface Integrals 443

10.7 Triple Integrals. Divergence Theorem of Gauss 452

10.8 Further Applications of the Divergence Theorem 458

10.9 Stokes’s Theorem 463

11.1 Fourier Series 474

11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483

11.3 Forced Oscillations 492

11.4 Approximation by Trigonometric Polynomials 495

11.5 Sturm–Liouville Problems. Orthogonal Functions 498

11.6 Orthogonal Series. Generalized Fourier Series 504

11.7 Fourier Integral 510

11.8 Fourier Cosine and Sine Transforms 518

11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522

11.10 Tables of Transforms 534

12.1 Basic Concepts of PDEs 540

12.2 Modeling: Vibrating String, Wave Equation 543

12.3 Solution by Separating Variables. Use of Fourier Series 545

12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 553

12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557

12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem 558

12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms 568

12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575

12.9 Rectangular Membrane. Double Fourier Series 577

12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series 585

12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 593

12.12 Solution of PDEs by Laplace Transforms 600

13.1 Complex Numbers and Their Geometric Representation 608

13.2 Polar Form of Complex Numbers. Powers and Roots 613

13.3 Derivative. Analytic Function 619

13.4 Cauchy–Riemann Equations. Laplace’s Equation 625

13.5 Exponential Function 630

13.6 Trigonometric and Hyperbolic Functions. Euler's Formula 633

13.7 Logarithm. General Power. Principal Value 636

14.1 Line Integral in the Complex Plane 643

14.2 Cauchy's Integral Theorem 652

14.3 Cauchy's Integral Formula 660

14.4 Derivatives of Analytic Functions 664

15.1 Sequences, Series, Convergence Tests 671

15.2 Power Series 680

15.3 Functions Given by Power Series 685

15.4 Taylor and Maclaurin Series 690

15.5 Uniform Convergence. Optional 698

16.1 Laurent Series 708

16.2 Singularities and Zeros. Infinity 714

16.3 Residue Integration Method 719

16.4 Residue Integration of Real Integrals 725

17.1 Geometry of Analytic Functions: Conformal Mapping 736

17.2 Linear Fractional Transformations (Möbius Transformations) 741

17.3 Special Linear Fractional Transformations 745

17.4 Conformal Mapping by Other Functions 749

17.5 Riemann Surfaces. Optional 753

18.1 Electrostatic Fields 757

18.2 Use of Conformal Mapping. Modeling 761

18.3 Heat Problems 765

18.4 Fluid Flow 768

18.5 Poisson's Integral Formula for Potentials 774

18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem 778

19.1 Introduction 788

19.2 Solution of Equations by Iteration 795

19.3 Interpolation 805

19.4 Spline Interpolation 817

19.5 Numeric Integration and Differentiation 824

20.1 Linear Systems: Gauss Elimination 841

20.2 Linear Systems: LU-Factorization, Matrix Inversion 849

20.3 Linear Systems: Solution by Iteration 855

20.4 Linear Systems: Ill-Conditioning, Norms 861

20.5 Least Squares Method 869

20.6 Matrix Eigenvalue Problems: Introduction 873

20.7 Inclusion of Matrix Eigenvalues 876

20.8 Power Method for Eigenvalues 882

20.9 Tridiagonalization and QR-Factorization 885

21.1 Methods for First-Order ODEs 898

21.2 Multistep Methods 908

21.3 Methods for Systems and Higher Order ODEs 912

21.4 Methods for Elliptic PDEs 919

21.5 Neumann and Mixed Problems. Irregular Boundary 928

21.6 Methods for Parabolic PDEs 933

21.7 Method for Hyperbolic PDEs 939

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 949

22.2 Linear Programming 952

22.3 Simplex Method 956

22.4 Simplex Method: Difficulties 960

23.1 Graphs and Digraphs 967

23.2 Shortest Path Problems. Complexity 972

23.3 Bellman's Principle. Dijkstra’s Algorithm 977

23.4 Shortest Spanning Trees: Greedy Algorithm 980

23.5 Shortest Spanning Trees: Prim’s Algorithm 984

23.6 Flows in Networks 987

23.7 Maximum Flow: Ford–Fulkerson Algorithm 993

23.8 Bipartite Graphs. Assignment Problems 996

APPENDIX 1 References A1

APPENDIX 2 Answers to Selected Problems A4

APPENDIX 3 Auxiliary Material A51

INDEX I1