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9780132637572

Differential Equations and Linear Algebra

by ;
  • ISBN13:

    9780132637572

  • ISBN10:

    013263757X

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2008-01-01
  • Publisher: Addison Wesley
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Supplemental Materials

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Summary

This very accessible guide offers a thorough introduction to the basics of differential equations and linear algebra. Expertly integrating the two topics, it explains concepts clearly and logically -without sacrificing level or rigor - and supports material with a vast array of problems of varying levels for readers to choose from. Promotes in-depth understanding (vs. rote memorization) - enabling readers to fully comprehend abstract concepts and finish with a solid and working knowledge of linear mathematics. Offers one of the most lucid and clearly written narratives on the subject, with material that is accessible to the average reader, yet challenging to all. Presents a greater emphasis on geometry to help users better visualize the abstract concepts, and illustrates all concepts with an ample amount of worked examples. Second Edition highlights include new discussions direction fields and Euler's method for first order differential equations; row space and column space of a matrix, and the rank-nullity theorem; non-linear systems of differential equations, including phase plane analysis; and change of variables for differential equations. Now features a chapter on second order linear differential equations that isnot based on vector space methods to gives users a firmer grasp of the differential equation concept early on, and also on the solution techniques for this important class of differential equations.

Table of Contents

Preface xiii
First-Order Differential Equations
1(105)
How Differential Equations Arise
1(8)
Basic Ideas and Terminology
9(9)
The Geometry of First-Order DE
18(11)
Separable DE
29(9)
Some Simple Population Models
38(6)
First-Order Linear DE
44(7)
Two Modeling Problems Governed by First-Order Linear DE
51(10)
Change of Variables
61(10)
Exact DE
71(9)
Summary of Techniques for Solving First-Order DE
80(2)
Numerical Solution to First-Order DE
82(9)
Some Higher Order DE
91(5)
The Phase Plane
96(10)
Second-Order Linear Differential Equations
106(61)
Basic Theoretical Results
107(7)
Reduction of Order
114(5)
Second-Order Homogeneous Constant Coefficient Linear DE
119(6)
The Method of Undetermined Coefficients
125(8)
Complex-Valued Trial Solutions
133(3)
Oscillations of a Mechanical System
136(14)
RLC Circuits
150(4)
The Variation-of-Parameters Method
154(6)
A DE with Nonconstant Coefficients
160(7)
Matrices and Systems of Linear Algebraic Equations
167(65)
Matrices: Definitions and Notation
168(4)
Matrix Algebra
172(13)
Terminology and Notation for Systems of Linear Equations
185(5)
Elementary Row Operations and Row-Echelon Matrices
190(11)
Gaussian Elimination
201(11)
The Inverse of a Square Matrix
212(10)
Elementary Matrices and the LU Factorization
222(10)
Determinants
232(39)
The Definition of a Determinant
232(10)
Properties of Determinants
242(11)
Cofactor Expansions
253(11)
Summary of Determinants
264(7)
Vector Spaces
271(84)
Vectors in Rn
272(6)
Definition of a Vector Space
278(7)
Subspaces
285(7)
Spanning Sets
292(8)
Linear Dependence and Linear Independence
300(12)
Bases and Dimension
312(12)
Row Space and Column Space
324(5)
The Rank-Nullity Theorem
329(5)
Inner Product Spaces
334(9)
Orthogonal Sets of Vectors and the Gram-Schmidt Procedure
343(8)
Summary of Results
351(4)
Linear Transformations and the Eigenvalue/Eigenvector Problem
355(67)
Definition of a Linear Transformation
356(8)
Transformations of R2
364(7)
The Kernel and Range of a Linear Transformation
371(7)
Further Properties of Linear Transformations
378(7)
The Algebraic Eigenvalue/Eigenvector Problem
385(11)
General Results for Eigenvalues and Eigenvectors
396(7)
Diagonalization
403(7)
Orthogonal Diagonalization and Quadratic Forms
410(10)
Summary of Results
420(2)
Linear Differential Equations of Order n
422(26)
General Theory for Linear Differential Equations
423(7)
Constant Coefficient Homogeneous Linear DE
430(5)
The Method of Undetermined Coefficients: Annihilators
435(7)
The Variation-of-Parameters Method
442(6)
Systems of Differential Equations
448(85)
Introduction
448(2)
First-Order Linear Systems
450(5)
Vector Formulation
455(6)
General Results for First-Order Linear Differential Systems
461(5)
Homogeneous Constant Coefficient VDE: Nondefective Coefficient Matrix
466(9)
Homogeneous Constant Coefficient VDE: Defective Coefficient Matrix
475(10)
Variation-of-Parameters for Linear Systems
485(5)
Some Applications of Linear Systems of Differential Equations
490(11)
An Introduction to the Matrix Exponential Function
501(5)
The Matrix Exponential Function and Systems of Differential Equations
506(8)
The Phase Plane for Linear Autonomous Systems
514(9)
Nonlinear Systems
523(10)
The Laplace Transform and Some Elementary Applications
533(46)
The Definition of the Laplace Transform
533(6)
The Existence of the Laplace Transform and the Inverse Transform
539(5)
Periodic Functions and the Laplace Transform
544(3)
The Transform of Derivatives and the Solution of Initial-Value Problems
547(5)
The First Shifting Theorem
552(4)
The Unit Step Function
556(3)
The Second Shifting Theorem
559(8)
Impulsive Driving Terms: The Dirac Delta Function
567(5)
The Convolution Integral
572(7)
Series Solutions to Differential Equations
579(60)
Introduction
579(1)
A Review of Power Series
580(7)
Series Solutions about an Ordinary Point
587(9)
The Legendre Equation
596(9)
Series Solutions about a Regular Singular Point
605(9)
Frobenius Theory
614(13)
Bessel's Equation of Order p
627(12)
Appendix 1 A Review of Complex Numbers 639(6)
Appendix 2 A Review of Partial Fractions 645(7)
Appendix 3 A Review of Integration Techniques 652(8)
Appendix 4 An Existence and Uniqueness Theorem for First-Order Differential Equations 660(12)
Appendix 5 Linearly Independent Solutions to x2y'' + xp(x)y' + q(x)y = 0 672(4)
Answers to Odd-Numbered Problems 676(23)
Index 699

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