Differential Equations and Linear Algebra, Digital Update, 4th edition

Stephen W. Goode
, Scott A. Annin

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Differential Equations & Linear Algebra, 4th Edition helps you develop an in-depth understanding versus rote memorization. It presents a carefully balanced and sound integration of both differential equations and linear algebra, enabling you to fully comprehend abstract concepts and leave the course with a solid foundation in key areas. Flexible in format, it explains concepts clearly and logically with an abundance of examples and illustrations, without sacrificing level or rigor. Many problems support the material, with varying difficulty levels from which students and instructors can choose. It is best suited for students who have successfully completed 3 semesters of calculus.

ISBN-13: 9780135314067

Subject: Advanced Math

Category: Differential Equations & Linear Alg (Combined)

1. First-Order Differential Equations

1.1 Differential Equations Everywhere
1.2 Basic Ideas and Terminology
1.3 The Geometry of First-Order Differential Equations
1.4 Separable Differential Equations
1.5 Some Simple Population Models
1.6 First-Order Linear Differential Equations
1.7 Modeling Problems Using First-Order Linear Differential Equations
1.8 Change of Variables
1.9 Exact Differential Equations
1.10 Numerical Solution to First-Order Differential Equations
1.11 Some Higher-Order Differential Equations
1.12 Chapter Review

2. Matrices and Systems of Linear Equations

2.1 Matrices: Definitions and Notation
2.2 Matrix Algebra
2.3 Terminology for Systems of Linear Equations
2.4 Row-Echelon Matrices and Elementary Row Operations
2.5 Gaussian Elimination
2.6 The Inverse of a Square Matrix
2.7 Elementary Matrices and the LU Factorization
2.8 The Invertible Matrix Theorem I
2.9 Chapter Review

3. Determinants

3.1 The Definition of the Determinant
3.2 Properties of Determinants
3.3 Cofactor Expansions
3.4 Summary of Determinants
3.5 Chapter Review

4. Vector Spaces

4.1 Vectors in Rn
4.2 Definition of a Vector Space
4.3 Subspaces
4.4 Spanning Sets
4.5 Linear Dependence and Linear Independence
4.6 Bases and Dimension
4.7 Change of Basis
4.8 Row Space and Column Space
4.9 The Rank-Nullity Theorem
4.10 Invertible Matrix Theorem II
4.11 Chapter Review

5. Inner Product Spaces

5.1 Definition of an Inner Product Space
5.2 Orthogonal Sets of Vectors and Orthogonal Projections
5.3 The Gram-Schmidt Process
5.4 Least Squares Approximation
5.5 Chapter Review

6. Linear Transformations

6.1 Definition of a Linear Transformation
6.2 Transformations of R2
6.3 The Kernel and Range of a Linear Transformation
6.4 Additional Properties of Linear Transformations
6.5 The Matrix of a Linear Transformation
6.6 Chapter Review

7. Eigenvalues and Eigenvectors

7.1 The Eigenvalue/Eigenvector Problem
7.2 General Results for Eigenvalues and Eigenvectors
7.3 Diagonalization
7.4 An Introduction to the Matrix Exponential Function
7.5 Orthogonal Diagonalization and Quadratic Forms
7.6 Jordan Canonical Forms
7.7 Chapter Review

8. Linear Differential Equations of Order n

8.1 General Theory for Linear Differential Equations
8.2 Constant Coefficient Homogeneous Linear Differential Equations
8.3 The Method of Undetermined Coefficients: Annihilators
8.4 Complex-Valued Trial Solutions
8.5 Oscillations of a Mechanical System
8.6 RLC Circuits
8.7 The Variation of Parameters Method
8.8 A Differential Equation with Nonconstant Coefficients
8.9 Reduction of Order
8.10 Chapter Review

9. Systems of Differential Equations

9.1 First-Order Linear Systems
9.2 Vector Formulation
9.3 General Results for First-Order Linear Differential Systems
9.4 Vector Differential Equations: Nondefective Coefficient Matrix
9.5 Vector Differential Equations: Defective Coefficient Matrix
9.6 Variation-Of-Parameters for Linear Systems
9.7 Some Applications of Linear Systems of Differential Equations
9.8 Matrix Exponential Function and Systems of Differential Equations
9.9 The Phase Plane for Linear Autonomous Systems
9.10 Nonlinear Systems
9.11 Chapter Review

10. The Laplace Transform and Some Elementary Applications

10.1 Definition of the Laplace Transform
10.2 The Existence of the Laplace Transform and the Inverse Transform
10.3 Periodic Functions and the Laplace Transform
10.4 The Transform of Derivatives and Solution of Initial-Value Problems
10.5 The First Shifting Theorem
10.6 The Unit Step Function
10.7 The Second Shifting Theorem
10.8 Impulsive Driving Terms: The Dirac Delta Function
10.9 The Convolution Integral
10.10 Chapter Review

11. Series Solutions to Linear Differential Equations

11.1 A Review of Power Series
11.2 Series Solutions About an Ordinary Point
11.3 The Legendre Equation
11.4 Series Solutions About a Regular Singular Point
11.5 Frobenius Theory
11.6 Bessel’s Equation of Order p
11.7 Chapter Review

Appendices

A. Review of Complex Numbers
B. Review of Partial Fractions
C. Review of Integration Techniques
D. Linearly Independent Solutions to x2y + xp(x)y + q(x)y = 0
E. Answers to Odd-Numbered Exercises

Index