Elementary Differential Equations with Boundary Value Problems with IDE CD Package

by Kohler, Werner E.; Johnson, Lee W.

• ISBN13: 9780321398505
• ISBN10: 0321398505
• Edition: 2nd
• Format: Hardcover w/CD
• Copyright: 2006-01-01
• Publisher: Addison Wesley

Summary

Elementary Differential Equations with Boundary Value Problems integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential equations, etc.) the text begins with the basic existence-uniqueness theory. This provides the student the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. The Table of Contents is comprehensive and allows flexibility for instructors.

Table of Contents

Preface	xi
Introduction to Differential Equations	1	(14)
Introduction	1	(1)
Examples of Differential Equations	1	(7)
Direction Fields	8	(7)
First Order Differential Equations	15	(92)
Introduction	15	(4)
First Order Linear Differential Equations	19	(12)
Introduction to Mathematical Models	31	(10)
Population Dynamics and Radioactive Decay	41	(7)
First Order Nonlinear Differential Equations	48	(6)
Separable First Order Equations	54	(9)
Exact Differential Equations	63	(7)
The Logistic Population Model	70	(7)
Applications to Mechanics	77	(12)
Euler's Method	89	(18)
Review Exercises	100	(1)
Projects	101	(6)
Second and Higher Order Linear Differential Equations	107	(106)
Introduction	108	(7)
The General Solution of Homogeneous Equations	115	(6)
Constant Coefficient Homogeneous Equations	121	(6)
Real Repeated Roots; Reduction of Order	127	(5)
Complex Roots	132	(10)
Unforced Mechanical Vibrations	142	(12)
The General Solution of a Linear Nonhomogeneous Equation	154	(4)
The Method of Undetermined Coefficients	158	(10)
The Method of Variation of Parameters	168	(6)
Forced Mechanical Vibrations, Electrical Networks, and Resonance	174	(14)
Higher Order Linear Homogeneous Differential Equations	188	(7)
Higher Order Homogeneous Constant Coefficient Differential Equations	195	(6)
Higher Order Linear Nonhomogeneous Differential Equations	201	(12)
Review Exercises	206	(1)
Projects	206	(7)
First Order Linear Systems	213	(104)
Introduction	213	(10)
Existence and Uniqueness	223	(5)
Homogeneous Linear Systems	228	(10)
Constant Coefficient Homogeneous Systems; the Eigenvalue Problem	238	(9)
Real Eigenvalues and the Phase Plane	247	(9)
Complex Eigenvalues	256	(10)
Repeated Eigenvalues	266	(11)
Nonhomogeneous Linear Systems	277	(11)
Numerical Methods for Systems of Linear Differential Equations	288	(12)
The Exponential Matrix and Diagonalization	300	(17)
Review Exercises	310	(1)
Projects	311	(6)
Laplace Transforms	317	(74)
Introduction	317	(12)
Laplace Transform Pairs	329	(15)
The Method of Partial Fractions	344	(6)
Laplace Transforms of Periodic Functions and System Transfer Functions	350	(9)
Solving Systems of Differential Equations	359	(9)
Convolution	368	(9)
The Delta Function and Impulse Response	377	(14)
Projects	385	(6)
Nonlinear Systems	391	(80)
Introduction	391	(9)
Equilibrium Solutions and Direction Fields	400	(13)
Conservative Systems	413	(11)
Stability	424	(9)
Linearization and the Local Picture	433	(15)
Two-Dimensional Linear Systems	448	(10)
Predator-Prey Population Models	458	(13)
Projects	466	(5)
Numerical Methods	471	(44)
Introduction	471	(2)
Euler's Method, Heun's Method, and the Modified Euler's Method	473	(6)
Taylor Series Methods	479	(14)
Runge-Kutta Methods	493	(22)
Appendix 1: Convergence of One-Step Methods	506	(1)
Appendix 2: Stability of One-Step Methods	507	(3)
Projects	510	(5)
Series Solutions of Linear Differential Equations	515	(50)
Introduction	515	(12)
Series Solutions Near an Ordinary Point	527	(9)
The Euler Equation	536	(6)
Solutions Near a Regular Singular Point and the Method of Frobenius	542	(8)
The Method of Frobenius Continued: Special Cases and a Summary	550	(15)
Projects	561	(4)
Second Order Partial Differential Equations and Fourier Series	565	(94)
Introduction	565	(5)
Heat Flow in a Thin Bar; Separation of Variables	570	(10)
Series Solutions	580	(9)
Calculating the Solution	589	(11)
Fourier Series	600	(16)
The Wave Equation	616	(12)
Laplace's Equation	628	(13)
Higher-Dimensional Problems; Nonhomogeneous Equations	641	(18)
Project	655	(4)
First Order Partial Differential Equations and the Method of Characteristics	659	(22)
Introduction	659	(3)
The Cauchy Problem	662	(6)
Existence and Uniqueness	668	(3)
The Method of Characteristics	671	(10)
Projects	679	(2)
Linear Two-Point Boundary Value Problems	681
Introduction	681	(1)
Existence and Uniqueness	682	(11)
Two-Point Boundary Value Problems for Linear Systems	693	(12)
Sturm-Liouville Boundary Value Problems	705
Project	715
Answers	1	(1)
Index	1