Elementary Differential Equations with Linear Algebra

PrefaceOne Introduction to Differential Equations 1.1 Introduction 1.2 Separable Equations 1.3 Exact Equations 1.4 First-Order Linear Equations 1.5 Orthogonal Trajectories 1.6 Radioactive Decay 1.7 Mixing Problems 1.8 Population Growth 1.9 Economic Models 1.10 Cooling; The Rate of a Chemical Reaction 1.11 Two Special Types of Second-Order Equations 1.12 Falling Bodies 1.13 Some Theoretical MattersTwo Matrices and Determinants 2.1 Systems of Linear Equations 2.2 Homogeneous Systems 2.3 Applications Involving Systems of Equations 2.4 Matrices and Vectors 2.5 Matrix Multiplication 2.6 Inner Product and Length 2.7 Some Special Matrices 2.8 Determinants 2.9 Properties of Determinants 2.10 Cofactors 2.11 Cramer's Rule 2.12 The Inverse of a MatrixThree Vector Spaces and Linear Transformations 3.1 Vector Spaces 3.2 Subspaces 3.3 Linear Dependence 3.4 Wronskians 3.5 Dimension 3.6 Orthogonal Bases 3.7 Linear Transformations 3.8 Properties of Linear Transformations 3.9 Differential OperatorsFour Characteristic Values 4.1 Characteristic Values 4.2 An Application: Population Growth 4.3 Diagonalization 4.4 Real Symmetric Matrices 4.5 Functions of MatricesFive Linear Differential Equations 5.1 Introduction 5.2 Polynomial Operators 5.3 Complex Solutions 5.4 Equations with Constant Coefficients 5.5 Cauchy-Euler Equations 5.6 Nonhomogeneous Equations 5.7 The Method of Undetermined Coefficients 5.8 Variation of Parameters 5.9 Simple Harmonic Motion 5.10 Electric CircuitsSix Systems of Differential Equations 6.1 Introduction 6.2 First-Order Systems 6.3 Linear Systems with Constant Coefficients 6.4 Matrix Formulation of Linear Systems 6.5 Fundamental Sets of Solutions 6.6 Solutions by Characteristic Values 6.7 Repeated Characteristic Values 6.8 Series of Matrices 6.9 The Exponential Matrix Function 6.10 A Matrix Method 6.11 Nonhomogeneous Linear Systems 6.12 Mechanical Systems 6.13 The Two-Body Problem 6.14 Electric Circuits 6.15 Some Problems from BiologySeven Series Solutions 7.1 Power Series 7.2 Taylor Series 7.3 Ordinary Points 7.4 Singular Points 7.5 The Case of Equal Exponents 7.6 The Case when the Exponents Differ by an Integer 7.7 The Point at Infinity 7.8 Legendre Polynomials 7.9 Bessel FunctionsEight Numerical Methods 8.1 The Euler Method 8.2 Taylor Series Methods 8.3 Runge-Kutta Methods 8.4 A Multi-Step MethodNine Laplace Transforms 9.1 The Laplace Transform 9.2 Functions of Exponential Order 9.3 Properties of Laplace Transforms 9.4 Inverse Transforms 9.5 Applications to Differential Equations 9.6 Functions with DiscontinuitiesReferencesAnswers to Selected ExercisesIndex