Elementary Differential Equations

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

• ISBN13: 9780201709261
• ISBN10: 0201709260
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-01-01
• 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 that provides students with the necessary framework to understand and solve differential equations. Theory is presented as simply as possible with an emphasis on how to use it. With an emphasis on linear equations, linear and nonlinear equations (first order and higher order) are treated in separate chapters. In developing mathematical models, this text guides the student carefully through the underlying physical principles leading to the relevant mathematics. Asking students to use common sense, intuition, and 'back-of-the-envelope' checks as well as challenging them to anticipate and interpret the physical content of the solution encourage critical thinking. MARKET: Intended for use in introductory course in differential equations.

Table of Contents

Preface	vii
Introduction to Differential Equations	1	(14)
Examples of Differential Equations	1	(7)
Direction Fields	8	(7)
First Order Linear Differential Equations	15	(44)
Existence and Uniqueness	16	(3)
First Order Linear Homogeneous Differential Equations	19	(8)
First Order Linear Nonhomogeneous Differential Equations	27	(8)
Introduction to Mathematical Models	35	(11)
Mixing Problems and Cooling Problems	46	(13)
The Processing of Seafood	56	(3)
First Order Nonlinear Differential Equations	59	(56)
Existence and Uniqueness	60	(5)
Separable First Order Equations	65	(9)
Exact Differential Equations	74	(7)
Bernoulli Equations	81	(2)
The Logistic Population Model	83	(7)
One-Dimensional Motion with Air Resistance	90	(5)
One-Dimensional Dynamics with Distance as the Independent Variable	95	(8)
Euler's Method	103	(12)
The Baranyi Population Model and the Storage of Whole Milk	111	(4)
Second Order Linear Differential Equations	115	(106)
Existence and Uniqueness	122	(3)
The General Solution of Homogeneous Equations	125	(10)
Fundamental Sets and Linear Independence	135	(9)
Constant Coefficient Homogeneous Equations	144	(9)
Real Repeated Roots; Reduction of Order	153	(6)
Complex Roots	159	(10)
Unforced Mechanical Vibrations	169	(14)
The General Solution of the Linear Nonhomogeneous Equation	183	(4)
The Method of Undetermined Coefficients	187	(11)
The Method of Variation of Parameters	198	(6)
Forced Mechanical Vibrations, Electrical Networks, and Resonance	204	(17)
Modeling Buoyant Motion	218	(3)
Higher Order Linear Differential Equations	221	(32)
Existence and Uniqueness	222	(1)
The General Solution of nth Order Linear Homogeneous Equations	223	(7)
Fundamental Sets and Linear Independence	230	(6)
Constant Coefficient Homogeneous Equations	236	(7)
Nonhomogeneous Linear Equations	243	(10)
The Loading of a Beam	250	(3)
First Order Linear Systems	253	(124)
The Calculus of Matrix Functions	256	(8)
Existence and Uniqueness	264	(6)
Homogeneous Linear Systems	270	(10)
Fundamental Sets and Linear Independence	280	(5)
Constant Coefficient Homogeneous Systems	285	(13)
Complex Eigenvalues	298	(13)
Repeated Eigenvalues	311	(12)
Nonhomogeneous Linear Systems	323	(14)
Euler's Method for Systems of Differential Equations	337	(10)
Diagonalization	347	(12)
Propagator Matrices, Functions of a Matrix, and the Exponential Matrix	359	(18)
Three Chemical Reactants	374	(3)
Laplace Transforms	377	(78)
The Laplace Transform	380	(10)
Laplace Transform Pairs	390	(16)
The Method of Partial Fractions	406	(7)
Laplace Transforms of Periodic Functions and System Transfer Functions	413	(11)
Solving Systems of Differential Equations	424	(10)
Convolution	434	(9)
The Delta Function and Impulse Response	443	(12)
Locating a Transmission Line Fault	451	(4)
Nonlinear Systems	455	(82)
Existence and Uniqueness	460	(6)
Equilibrium Solutions and Direction Fields	466	(13)
Conservative Systems	479	(11)
Stability	490	(10)
Linearization and the Local Picture	500	(15)
Two-Dimensional Linear Systems	515	(10)
Predator-Prey Population Models	525	(12)
Introduction of an Infectious Disease into a Predator-Prey Ecosystem	535	(2)
Numerical Methods	537	(42)
Euler's Method, Heun's Method, and the Modified Euler's Method	539	(7)
Taylor Series Methods	546	(13)
Runge-Kutta Methods	559	(20)
Projectile Motion	576	(3)
Series Solutions of Linear Differential Equations	579	(58)
Review of Power Series	581	(11)
Series Solutions Near an Ordinary Point	592	(12)
The Euler Equation	604	(6)
Solutions Near a Regular Singular Point and the Method of Frobenius	610	(9)
The Method of Frobenius Continued: Special Cases and a Summary	619	(18)
Steady-State Heat Flow between Concentric Cylinders	631	(6)
Appendix: Review of Matrices and Vectors	637
Answers to Odd-Numbered Exercises	1	(1)
Index	1