Elementary Differential Equations With Boundary Value Problems

by Edwards, C. H.; Penney, David E.

• ISBN13: 9780130113016
• ISBN10: 0130113018
• Edition: 4th
• Format: Hardcover
• Copyright: 1999-10-01
• Publisher: PRENTICE HALL

Summary

For courses in differential equations. This book, with enough material for 2 terms, provides a concrete and readable text for the traditional course in elementary differential equations that science, engineering, and mathematics students take following calculus. This is a strongly algebraic-oriented text with some computer enhancements for numerical methods. Matters of definition, classification, and logical structure deserve (and receive here) careful attention for the first time in the mathematical experience of many of the students. While it is neither feasible nor desirable to include proofs of the fundamental existence and uniqueness theorems along the way in an elementary course, students need to see precise and clear-cut statements of these theorems, and understand their role in the subject. Appropriate existence and uniqueness proofs in the Appendix are included, and referred to where appropriate in the main body of the text.

Table of Contents

Preface	ix
First-Order Differential Equations	1	(93)
Differential Equations and Mathematical Models	1	(9)
Integrals as General and Particular Solutions	10	(8)
Direction Fields and Solution Curves	18	(12)
Separable Equations and Applications	30	(13)
Linear First-Order Equations	43	(13)
Substitution Methods and Exact Equations	56	(14)
Population Models	70	(11)
Acceleration-Velocity Models	81	(13)
Linear Equations of Higher Order	94	(97)
Introduction: Second-Order Linear Equations	94	(15)
General Solutions of Linear Equations	109	(12)
Homogeneous Equations with Constant Coefficients	121	(11)
Mechanical Vibrations	132	(12)
Undetermined Coefficients and Variation of Parameters	144	(14)
Forced Oscillations and Resonance	158	(12)
Electrical Circuits	170	(7)
Endpoint Problems and Eigenvalues	177	(14)
Power Series Methods	191	(72)
Introduction and Review of Power Series	191	(14)
Series Solutions Near Ordinary Points	205	(11)
Regular Singular Points	216	(15)
Method of Frobenius: The Exceptional Cases	231	(16)
Bessel's Equation	247	(10)
Applications of Bessel Functions	257	(6)
Laplace Transform Methods	263	(61)
Laplace Transforms and Inverse Transforms	263	(11)
Transformation of Initial Value Problems	274	(9)
Translation and Partial Fractions	283	(9)
Derivatives, Integrals, and Products of Transforms	292	(8)
Periodic and Piecewise Continuous Input Functions	300	(13)
Impulses and Delta Functions	313	(11)
Linear Systems of Differential Equations	324	(111)
First-Order Systems and Applications	324	(14)
The Method of Elimination	338	(10)
Matrices and Linear Systems	348	(20)
The Eigenvalue Method for Homogeneous Systems	368	(16)
Second-Order Systems and Mechanical Applications	384	(13)
Multiple Eigenvalue Solutions	397	(16)
Matrix Exponentials and Linear Systems	413	(13)
Nonhomogeneous Linear Systems	426	(9)
Numerical Methods	435	(51)
Numerical Approximation: Euler's Method	435	(11)
A Closer Look at the Euler Method	446	(11)
The Runge-Kutta Method	457	(12)
Numerical Methods for Systems	469	(17)
Nonlinear Systems and Phenomena	486	(72)
Equilibrium Solutions and Stability	486	(7)
Stability and the Phase Plane	493	(13)
Linear and Almost Linear Systems	506	(13)
Ecological Models: Predators and Competitors	519	(13)
Nonlinear Mechanical Systems	532	(13)
Chaos in Dynamical Systems	545	(13)
Fourier Series Methods	558	(74)
Periodic Functions and Trigonometric Series	558	(9)
General Fourier Series and Convergence	567	(8)
Fourier Sine and Cosine Series	575	(11)
Applications of Fourier Series	586	(6)
Heat Conduction and Separation of Variables	592	(16)
Vibrating Strings and the One-Dimensional Wave Equation	608	(13)
Steady-State Temperature and Laplace's Equation	621	(11)
Eigenvalues and Boundary Value Problems	632	(66)
Sturm-Liouville Problems and Eigenfunction Expansions	632	(12)
Applications of Eigenfunction Series	644	(11)
Steady Periodic Solutions and Natural Frequencies	655	(8)
Cylindrical Coordinate Problems	663	(15)
Higher-Dimensional Phenomena	678	(20)
References for Further Study	698	(2)
Appendix: Existence and Uniqueness of Solutions	700	(16)
Answers to Selected Problems	716	(21)
Index	737