Differential Equations and Linear Algebra

by Goode, Stephen W.; Annin, Scott A.

• ISBN13: 9780132637572
• ISBN10: 013263757X
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: Addison Wesley

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