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ELEMENTARY DIFFERENTIAL EQUATIONS WITH …

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. This book has been judged to meet the evaluation criteria set by the Edi- torial Board of the American Institute of Mathematics in connection with the Institute's Open Textbook Initiative. It may be copied, modified, re- distributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike Unported License.

Section 13.1 deals with two-point value problems for a second order ordinary differential equation. Conditionsfor existence and uniquenessof solutionsare given, andthe constructionofGreen’s functions is included. Section 13.2 presents the elementary aspects of Sturm-Liouvilletheory. You may also find the followingto be of interest:

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Transcription of ELEMENTARY DIFFERENTIAL EQUATIONS WITH …

1 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. This book has been judged to meet the evaluation criteria set by the Edi- torial Board of the American Institute of Mathematics in connection with the Institute's Open Textbook Initiative. It may be copied, modified, re- distributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike Unported License.

2 FREE DOWNLOAD: STUDENT SOLUTIONS MANUAL. Free Edition (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. Applications Leading to DIFFERENTIAL EQUATIONS First Order EQUATIONS 5.

3 Direction Fields for First Order EQUATIONS 16. Chapter 2 First Order EQUATIONS 30. Linear First Order EQUATIONS 30. Separable EQUATIONS 45. Existence and Uniqueness of Solutions of Nonlinear EQUATIONS 55. Transformation of Nonlinear EQUATIONS into Separable EQUATIONS 62. Exact EQUATIONS 73. Integrating Factors 82. Chapter 3 Numerical Methods Euler's Method 96. The Improved Euler Method and Related Methods 109. The Runge-Kutta Method 119. Chapter 4 Applications of First Order Equations1em 130. Growth and Decay 130.

4 Cooling and Mixing 140. ELEMENTARY Mechanics 151. Autonomous Second Order EQUATIONS 162. Applications to Curves 179. Chapter 5 Linear Second Order EQUATIONS Homogeneous Linear EQUATIONS 194. Constant Coefficient Homogeneous EQUATIONS 210. Nonhomgeneous Linear EQUATIONS 221. The Method of Undetermined Coefficients I 229. iv The Method of Undetermined Coefficients II 238. Reduction of Order 248. Variation of Parameters 255. Chapter 6 Applcations of Linear Second Order EQUATIONS 268. Spring Problems I 268.

5 Spring Problems II 279. The RLC Circuit 290. Motion Under a Central Force 296. Chapter 7 Series Solutions of Linear Second Order EQUATIONS Review of Power Series 306. Series Solutions Near an ordinary Point I 319. Series Solutions Near an ordinary Point II 334. Regular Singular Points Euler EQUATIONS 342. The Method of Frobenius I 347. The Method of Frobenius II 364. The Method of Frobenius III 378. Chapter 8 Laplace Transforms Introduction to the Laplace Transform 393. The Inverse Laplace Transform 405.

6 Solution of Initial Value Problems 413. The Unit Step Function 419. Constant Coefficient EQUATIONS with Piecewise Continuous Forcing Functions 430. Convolution 440. Constant Cofficient EQUATIONS with Impulses 452. A Brief Table of Laplace Transforms Chapter 9 Linear Higher Order EQUATIONS Introduction to Linear Higher Order EQUATIONS 465. Higher Order Constant Coefficient Homogeneous EQUATIONS 475. Undetermined Coefficients for Higher Order EQUATIONS 487. Variation of Parameters for Higher Order EQUATIONS 497.

7 Chapter 10 Linear Systems of DIFFERENTIAL EQUATIONS Introduction to Systems of DIFFERENTIAL EQUATIONS 507. Linear Systems of DIFFERENTIAL EQUATIONS 515. Basic Theory of Homogeneous Linear Systems 521. Constant Coefficient Homogeneous Systems I 529. vi Contents Constant Coefficient Homogeneous Systems II 542. Constant Coefficient Homogeneous Systems II 556. Variation of Parameters for Nonhomogeneous Linear Systems 568. Chapter 11 Boundary Value Problems and Fourier Expansions 580. Eigenvalue Problems for y00 + y = 0 580.

8 Fourier Series I 586. Fourier Series II 603. Chapter 12 Fourier Solutions of Partial DIFFERENTIAL EQUATIONS The Heat Equation 618. The Wave Equation 630. Laplace's Equation in Rectangular Coordinates 649. Laplace's Equation in Polar Coordinates 666. Chapter 13 Boundary Value Problems for Second Order Linear EQUATIONS Boundary Value Problems 676. Sturm Liouville Problems 687. Preface ELEMENTARY DIFFERENTIAL EQUATIONS with Boundary Value Problems is written for students in science, en- gineering, and mathematics who have completed calculus through partial differentiation.

9 If your syllabus includes Chapter 10 (Linear Systems of DIFFERENTIAL EQUATIONS ), your students should have some prepa- ration in linear algebra. In writing this book I have been guided by the these principles: An ELEMENTARY text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student's place, and have chosen to err on the side of too much detail rather than not enough. An ELEMENTARY text can't be better than its exercises. This text includes 2041 numbered exercises, many with several parts.

10 They range in difficulty from routine to very challenging. An ELEMENTARY text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language that students can understand. I have minimized the number of explicitly stated theorems and def- initions, preferring to deal with concepts in a more conversational way, copiously illustrated by 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188.


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