Elementary Differential Equations with Linear Algebra

by Rabenstein, Albert L.

• ISBN13: 9780155209848
• ISBN10: 0155209841
• Edition: 4th
• Format: Hardcover
• Copyright: 1992-01-02
• Publisher: Cengage Learning

Summary

Designed for use by sophomore engineering or junior physical science majors, this text is suitable for an introductory course in linear algebra and differential equations or a course in differential equations with a linear algebra prerequisite. This text contains detailed coverage of applied topics and includes theorems specifically applicable to engineering students. There is a new chapter on "Stability and the Phase Plane", approximately 300 new problems added throughout and several BASIC programs on numerical solutions of differential equations are included.

Table of Contents

Introduction to Differential Equations
Introduction	1	(7)
Separable Equations	8	(4)
Homogeneous Equations	12	(5)
Exact Equations	17	(6)
First-Order Linear Equations	23	(5)
Orthogonal Trajectories	28	(4)
Radioactive Decay	32	(3)
Mixing Problems	35	(2)
Population Growth	37	(2)
Cooling; The Rate of a Chemical Reaction	39	(5)
Two Special Types of Second-Order Equations	44	(4)
Falling Bodies	48	(7)
Some Theoretical Matters	55	(8)
Matrices and Determinants
Systems of Linear Equations	63	(12)
Homogeneous Systems	75	(4)
Applications	79	(3)
Matrices and Vectors	82	(5)
Matrix Multiplication	87	(6)
Inner Product and Length	93	(4)
Some Special Matrices	97	(6)
Determinants	103	(4)
Properties of Determinants	107	(5)
Cofactors	112	(3)
Cramer's Rule	115	(6)
The Inverse of a Matrix	121	(8)
Vector Spaces and Linear Transformations
Vector Spaces	129	(4)
Subspaces	133	(4)
Linear Dependence	137	(5)
Wronskians	142	(4)
Dimension	146	(4)
Orthogonal Bases	150	(5)
Linear Transformations	155	(5)
Properties of Linear Transformations	160	(3)
Transformations of the Plane	163	(6)
Differential Operators	169	(8)
Characteristic Values
Characteristic Values	177	(5)
An Application	182	(2)
Diagonalization	184	(8)
Real Symmetric Matrices	192	(5)
Functions of Matrices	197	(6)
Linear Differential Equations
Introduction	203	(4)
Polynomial Operators	207	(4)
Complex Solutions	211	(7)
Equations with Constant Coefficients	218	(7)
Cauchy-Euler Equations	225	(4)
Nonhomogeneous Equations	229	(3)
The Method of Undetermined Coefficients	232	(9)
Variation of Parameters	241	(8)
Simple Harmonic Motion	249	(8)
Electric Circuits	257	(6)
Systems of Differential Equations
Introduction	263	(4)
First-Order Systems	267	(4)
Linear Systems with Constant Coefficients	271	(8)
Matrix Formulation of Linear Systems	279	(4)
Fundamental Sets of Solutions	283	(5)
Solutions by Characteristic Values	288	(4)
Repeated Characteristic Values	292	(5)
Series of Matrices	297	(5)
The Exponential Matrix Function	302	(4)
A Matrix Method	306	(7)
Nonhomogeneous Linear Systems	313	(4)
Mechanical Systems	317	(5)
The Two Body Problem	322	(5)
Electric Circuits	327	(6)
Series Solutions
Power Series	333	(5)
Taylor Series	338	(3)
Ordinary Points	341	(6)
Singular Points	347	(8)
The Case of Equal Exponents	355	(5)
The Case When the Exponents Differ by an Integer	360	(5)
The Point at Infinity	365	(2)
Legendre Polynomials	367	(4)
Bessel Functions	371	(10)
Numerical Methods
The Euler Method	381	(5)
Taylor Series Methods	386	(3)
Runge-Kutta Methods	389	(4)
A Multi-Step Method	393	(5)
Systems of Equations	398	(7)
Laplace Transforms
The Laplace Transform	405	(4)
Functions of Exponential Order	409	(4)
Properties of Laplace Transforms	413	(4)
Inverse Transforms	417	(5)
Applications to Differential Equations	422	(5)
Functions with Discontinuities	427	(10)
Stability and the Phase Plane
Basic Ideas	437	(7)
The Phase Plane	444	(9)
Nonlinear Systems	453	(7)
Competition Between Two Species	460	(12)
Periodic Solutions	472	(7)
References	479	(2)
Answers to Selected Exercises	481	(26)
Index	507