ELEMENTARY DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS
ELEMENTARY DIFFERENTIAL EQUATIONS WITH
BOUNDARY VALUE PROBLEMS
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus Department of Mathematics Trinity University San Antonio, Texas, USA wtrench@trinity.edu
This book has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute's Open Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
FREE DOWNLOAD: STUDENT SOLUTIONS MANUAL
Free Edition 1.01 (December 2013)
This book was published previously by Brooks/Cole Thomson Learning, 2001. This free edition is made available in the hope that it will be useful as a textbook or reference. Reproduction is permitted for any valid noncommercial educational, mathematical, or scientific purpose. However, charges for profit beyond reasonable printing costs are prohibited.
TO BEVERLY
Contents
Chapter 1 Introduction
1
1.1 Applications Leading to Differential Equations
1.2 First Order Equations
5
1.3 Direction Fields for First Order Equations
16
Chapter 2 First Order Equations
30
2.1 Linear First Order Equations
30
2.2 Separable Equations
45
2.3 Existence and Uniqueness of Solutions of Nonlinear Equations
55
2.4 Transformation of Nonlinear Equations into Separable Equations
62
2.5 Exact Equations
73
2.6 Integrating Factors
82
Chapter 3 Numerical Methods
3.1 Euler's Method
96
3.2 The Improved Euler Method and Related Methods
109
3.3 The Runge-Kutta Method
119
Chapter 4 Applications of First Order Equations1em
130
4.1 Growth and Decay
130
4.2 Cooling and Mixing
140
4.3 Elementary Mechanics
151
4.4 Autonomous Second Order Equations
162
4.5 Applications to Curves
179
Chapter 5 Linear Second Order Equations
5.1 Homogeneous Linear Equations
194
5.2 Constant Coefficient Homogeneous Equations
210
5.3 Nonhomgeneous Linear Equations
221
5.4 The Method of Undetermined Coefficients I
229
iv
5.5 The Method of Undetermined Coefficients II
238
5.6 Reduction of Order
248
5.7 Variation of Parameters
255
Chapter 6 Applcations of Linear Second Order Equations
268
6.1 Spring Problems I
268
6.2 Spring Problems II
279
6.3 The RLC Circuit
290
6.4 Motion Under a Central Force
296
Chapter 7 Series Solutions of Linear Second Order Equations
7.1 Review of Power Series
306
7.2 Series Solutions Near an Ordinary Point I
319
7.3 Series Solutions Near an Ordinary Point II
334
7.4 Regular Singular Points Euler Equations
342
7.5 The Method of Frobenius I
347
7.6 The Method of Frobenius II
364
7.7 The Method of Frobenius III
378
Chapter 8 Laplace Transforms
8.1 Introduction to the Laplace Transform
393
8.2 The Inverse Laplace Transform
405
8.3 Solution of Initial Value Problems
413
8.4 The Unit Step Function
419
8.5 Constant Coefficient Equations with Piecewise Continuous Forcing
Functions
430
8.6 Convolution
440
8.7 Constant Cofficient Equations with Impulses
452
8.8 A Brief Table of Laplace Transforms
Chapter 9 Linear Higher Order Equations
9.1 Introduction to Linear Higher Order Equations
465
9.2 Higher Order Constant Coefficient Homogeneous Equations
475
9.3 Undetermined Coefficients for Higher Order Equations
487
9.4 Variation of Parameters for Higher Order Equations
497
Chapter 10 Linear Systems of Differential Equations
10.1 Introduction to Systems of Differential Equations
507
10.2 Linear Systems of Differential Equations
515
10.3 Basic Theory of Homogeneous Linear Systems
521
10.4 Constant Coefficient Homogeneous Systems I
529
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