Elementary Differential Equations with Linear Algebra (eBook)

Front Cover 1
Elementary Differential Equations with Linear Algebra 4
Copyright Page 5
Table of Contents 6
Preface 10
Chapter 1. Introduction to Differential Equations 12
1.1 Introduction 12
1.2 Separable Equations 20
1.3 Exact Equations 28
1.4 First-Order Linear Equations 35
1.5 Orthogonal Trajectories 40
1.6 Radioactive Decay 44
1.7 Mixing Problems 48
1.8 Population Growth 50
1.9 Economic Models 52
1.10 Cooling The Rate of a Chemical Reaction
1.11 Two Special Types of Second-Order Equations 61
1.12 Falling Bodies 65
1.13 Some Theoretical Matters 74
Chapter 2. Matrices and Determinants 84
2.1 Systems of Linear Equations 84
2.2 Homogeneous Systems 97
2.3 Applications Involving Systems of Equations 101
2.4 Matrices and Vectors 106
2.5 Matrix Multiplication 111
2.6 Inner Product and Length 118
2.7 Some Special Matrices 123
2.8 Determinants 128
2.9 Properties of Determinants 133
2.10 Cofactors 139
2.11 Cramer's Rule 142
2.12 The Inverse of a Matrix 149
Chapter 3. Vector Spaces and Linear Transformations 158
3.1 Vector Spaces 158
3.2 Subspaces 163
3.3 Linear Dependence 167
3.4 Wronskians 172
3.5 Dimension 178
3.6 Orthogonal Bases 181
3.7 Linear Transformations 185
3.8 Properties of Linear Transformations 189
3.9 Differential Operators 193
Chapter 4. Characteristic Values 202
4.1 Characteristic Values 202
4.2 An Application: Population Growth 207
4.3 Diagonalization 210
4.4 Real Symmetric Matrices 214
4.5 Functions of Matrices 217
Chapter 5. Linear Differential Equations 224
5.1 Introduction 224
5.2 Polynomial Operators 229
5.3 Complex Solutions 235
5.4 Equations with Constant Coefficients 241
5.5 Cauchy-Euler Equations 247
5.6 Nonhomogeneous Equations 253
5.7 The Method of Undetermined Coefficients 256
5.8 Variation of Parameters 267
5.9 Simple Harmonic Motion 276
5.10 Electric Circuits 285
Chapter 6. Systems of Differential Equations 294
6.1 Introduction 294
6.2 First-Order Systems 299
6.3 Linear Systems with Constant Coefficients 303
6.4 Matrix Formulation of Linear Systems 312
6.5 Fundamental Sets of Solutions 317
6.6 Solutions by Characteristic Values 324
6.7 Repeated Characteristic Values 328
6.8 Series of Matrices 332
6.9 The Exponential Matrix Function 338
6.10 A Matrix Method 343
6.11 Nonhomogeneous Linear Systems 351
6.12 Mechanical Systems 357
6.13 The Two-Body Problem 362
6.14 Electric Circuits 369
6.15 Some Problems from Biology 372
Chapter 7. Series Solutions 378
7.1 Power Series 378
7.2 Taylor Series 384
7.3 Ordinary Points 387
7.4 Singular Points 394
7.5 The Case of Equal Exponents 402
7.6 The Case when the Exponents Differ by an Integer 407
7.7 The Point at Infinity 413
7.8 Legendre Polynomials 415
7.9 Bessel Functions 420
Chapter 8. Numerical Methods 430
8.1 The Euler Method 430
8.2 Taylor Series Methods 437
8.3 Runge-Kutta Methods 441
8.4 A Multi-Step Method 445
Chapter 9. Laplace Transforms 454
9.1 The Laplace Transform 454
9.2 Functions of Exponential Order 459
9.3 Properties of Laplace Transforms 463
9.4 Inverse Transforms 468
9.5 Applications to Differential Equations 473
9.6 Functions with Discontinuities 479
References 492
Answers to Selected Exercises 494
Index 526