Elementary Differential Equations with Linear Algebra, Third Edition provides an introduction to differential equation and linear algebra. This book includes topics on numerical methods and Laplace transforms. Organized into nine chapters, this edition begins with an overview of an equation that involves a single unknown function of a single variable and some finite number of its derivatives. This text then examines a linear system of two equations with two unknowns. Other chapters consider a class of linear transformations that are defined on spaces of functions wherein these transformations are essential in the study of linear differential equations. This book discusses as well the linear differential equations whose coefficients are constant functions. The final chapter deals with the properties of Laplace transform in detail and examine as well the applications of Laplace transforms to differential equations. This book is a valuable resource for mathematicians, students, and research workers.
Inhalt
Preface
One 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 Matters
Two 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 Matrix
Three 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 Operators
Four Characteristic Values
4.1 Characteristic Values
4.2 An Application: Population Growth
4.3 Diagonalization
4.4 Real Symmetric Matrices
4.5 Functions of Matrices
Five 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 Circuits
Six 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 Biology
Seven 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 Functions
Eight Numerical Methods
8.1 The Euler Method
8.2 Taylor Series Methods
8.3 Runge-Kutta Methods
8.4 A Multi-Step Method
Nine 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 Discontinuities
References
Answers to Selected Exercises
Index