Differential Equations and Linear Algebra

Preface ix 1 First-Order Differential Equations 1 1.1 Differential Equations Everywhere 1 1.2 Basic Ideas and Terminology 13 1.3 The Geometry of First-Order Differential Equations 23 1.4 Separable Differential Equations 34 1.5 Some Simple Population Models 45 1.6 First-Order Linear Differential Equations 53 1.7 Modeling Problems Using First-Order Linear Differential Equations 61 1.8 Change of Variables 71 1.9 Exact Differential Equations 82 1.10 Numerical Solution to First-Order Differential Equations 93 1.11 Some Higher-Order Differential Equations 101 1.12 Chapter Review 106 2 Matrices and Systems of Linear Equations 114 2.1 Matrices: Definitions and Notation 115 2.2 Matrix Algebra 122 2.3 Terminology for Systems of Linear Equations 138 2.4 Row-Echelon Matrices and Elementary Row Operations 146 2.5 Gaussian Elimination 156 2.6 The Inverse of a Square Matrix 168 2.7 Elementary Matrices and the LU Factorization 179 2.8 The Invertible Matrix Theorem I 188 2.9 Chapter Review 190 3 Determinants 196 3.1 The Definition of the Determinant 196 3.2 Properties of Determinants 209 3.3 Cofactor Expansions 222 3.4 Summary of Determinants 235 3.5 Chapter Review 242 4 Vector Spaces 246 4.1 Vectors in Rn 248 4.2 Definition of a Vector Space 252 4.3 Subspaces 263 4.4 Spanning Sets 274 4.5 Linear Dependence and Linear Independence 284 4.6 Bases and Dimension 298 4.7 Change of Basis 311 4.8 Row Space and Column Space 319 4.9 The Rank-Nullity Theorem 325 4.10 Invertible Matrix Theorem II 331 4.11 Chapter Review 332 5 Inner Product Spaces 339 5.1 Definition of an Inner Product Space 340 5.2 Orthogonal Sets of Vectors and Orthogonal Projections 352 5.3 The Gram-Schmidt Process 362 5.4 Least Squares Approximation 366 5.5 Chapter Review 376 6 Linear Transformations 379 6.1 Definition of a Linear Transformation 380 6.2 Transformations of R2 391 6.3 The Kernel and Range of a Linear Transformation 397 6.4 Additional Properties of Linear Transformations 407 6.5 The Matrix of a Linear Transformation 419 6.6 Chapter Review 428 7 Eigenvalues and Eigenvectors 433 7.1 The Eigenvalue/Eigenvector Problem 434 7.2 General Results for Eigenvalues and Eigenvectors 446 7.3 Diagonalization 454 7.4 An Introduction to the Matrix Exponential Function 462 7.5 Orthogonal Diagonalization and Quadratic Forms 466 7.6 Jordan Canonical Forms 475 7.7 Chapter Review 488 8 Linear Differential Equations of Order n 493 8.1 General Theory for Linear Differential Equations 495 8.2 Constant Coefficient Homogeneous Linear Differential Equations 505 8.3 The Method of Undetermined Coefficients: Annihilators 515 8.4 Complex-Valued Trial Solutions 526 8.5 Oscillations of a Mechanical System 529 8.6 RLC Circuits 542 8.7 The Variation of Parameters Method 547 8.8 A Differential Equation with Nonconstant Coefficients 557 8.9 Reduction of Order 568 8.10 Chapter Review 573 9 Systems of Differential Equations 580 9.1 First-Order Linear Systems 582 9.2 Vector Formulation 588 9.3 General Results for First-Order Linear Differential Systems 593 9.4 Vector Differential Equations: Nondefective Coefficient Matrix 599 9.5 Vector Differential Equations: Defective Coefficient Matrix 608 9.6 Variation-Of-Parameters for Linear Systems 620 9.7 Some Applications of Linear Systems of Differential Equations 625 9.8 Matrix Exponential Function and Systems of Differential Equations 635 9.9 The Phase Plane for Linear Autonomous Systems 643 9.10 Nonlinear Systems 655 9.11 Chapter Review 663 10 The Laplace Transform and Some Elementary Applications 670 10.1 Definition of the Laplace Transform 670 10.2 The Existence of the Laplace Transform and the Inverse Transform 676 10.3 Periodic Functions and the Laplace Transform 682 10.4 The Transform of Derivatives and Solution of Initial-Value Problems 685 10.5 The First Shifting Theorem 690 10.6 The Unit Step Function 695 10.7 The Second Shifting Theorem 699 10.8 Impulsive Driving Terms: The Dirac Delta Function 706 10.9 The Convolution Integral 711 10.10 Chapter Review 717 11 Series Solutions to Linear Differential Equations 722 11.1 A Review of Power Series 723 11.2 Series Solutions About an Ordinary Point 731 11.3 The Legendre Equation 741 11.4 Series Solutions About a Regular Singular Point 750 11.5 Frobenius Theory 759 11.6 Bessel's Equation of Order p 773 11.7 Chapter Review 785
A Review of Complex Numbers 791 B Review of Partial Fractions 797 C Review of Integration Techniques 804 D Linearly Independent Solutions to x2y + xp(x)y + q(x)y = 0 811 E Answers to Odd-Numbered Exercises 000 Index