Schaums outlines differential equations 4th edition

Chapter 1 Basic Concepts 1Differential Equations 1 Notation 2 Solutions 2 Initial-Value and Boundary-Value Problems 2 Mathematical Models 9 The “Modeling Cycle” 9 Qualitative Methods 10

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Differential Equations

Equations

Fourth Edition

Richard Bronson, PhD

Professor of Mathematics and Computer Science

Fairleigh Dickinson University

Gabriel B Costa, PhD

Professor of Mathematical Sciences United States Military Academy Associate Professor of Mathematics and Computer Science

Seton Hall University

Schaum’s Outline Series

New York Chicago San Francisco Athens London Madrid Mexico City Milan New Delhi Singapore Sydney Toronto

simulation at the Technion–Israel Institute of Technology and the Wharton School of Business at the University of Pennsylvania Dr Bronson

has published over 30 technical articles and books, the latter including x Schaum’s Outline of Matrix Operations and Schaum’s Outline of

Operations Research.

GABRIEL B COSTA, PhD is a Catholic priest and Professor of Mathematical Sciences at the United States d Military Academy, West

Point, NY, where he also functions as an associate chaplain Father Costa is on extended leave from Seton Hall University, South Orange,

NJ He received his PhD in the area of differential equations from Stevens Institute of Technology in 1984 In addition to differential equations, Father Costa’s academic interests include mathematics education and sabermetrics, the search for objective knowledge about baseball

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To Ignace and Gwendolyn Bronson, Samuel and Rose Feldschuh — RB

To the great mathematicians and educators I have been blessed to meet: Professors Bloom, Brady, Bronson, Dostal, Goldfarb, Levine, Manogue, Pinkham, Pollara and Suffel …and, of course, Mr Rod! — GBC

Differential equations are among the linchpins of modern mathematics which, along with matrices, are essential for analyzing and solving complex problems in engineering, the natural sciences, econom- ics, and even business The emergence of low-cost, high-speed computers has spawned new tech- niques for solving differential equations, which allows problem solvers to model and solve complex problems based on systems of differential equations.

As with the two previous editions, this book outlines both the classical theory of differential tions and a myriad of solution techniques, including matrices, series methods, Laplace transforms and several numerical methods We have added a chapter on modeling and touch upon some qualitative methods that can be used when analytical solutions are difficult to obtain A chapter on classical dif- ferential equations (e.g., the equations of Hermite, Legendre, etc.) has been added to give the reader exposure to this rich, historical area of mathematics.

equa-This edition also features a chapter on difference equations and parallels this with differential equations Furthermore, we give the reader an introduction to partial differential equations and the solution techniques of basic integration and separation of variables Finally, we include an appendix

dealing with technology touching upon the TI-89 hand-held calculator and the MATHEMATICA

software packages.

With regard to both solved and supplementary problems, we have added such topics as integral equations of convolution type, Fibonacci numbers, harmonic functions, the heat equation and the wave equation We have also alluded to both orthogonality and weight functions with respect to classical differential equations and their polynomial solutions We have retained the emphasis on both initial value problems and differential equations without subsidiary conditions It is our aim to touch upon virtually every type of problem the student might encounter in a one-semester course on differential equations.

Each chapter of the book is divided into three parts The first outlines salient points of the theory and concisely summarizes solution procedures, drawing attention to potential difficulties and sub- tleties that too easily can be overlooked The second part consists of worked-out problems to clarify and,

on occasion, to augment the material presented in the first part Finally, there is a section of problems with answers that readers can use to test their understanding of the material.

The authors would like to thank the following individuals for their support and invaluable tance regarding this book We could not have moved as expeditiously as we did without their support and encouragement We are particularly indebted to Dean John Snyder and Dr Alfredo Tan of Fairleigh Dickinson University The continued support of the Most Reverend John J Myers, J.C.D., D.D., Archbishop of Newark, N.J., is also acknowledged From Seton Hall University we are grateful

assis-to the Reverend Monsignor James M Cafone and assis-to the members of the Priest Community; we also thank Dr Fredrick Travis, Dr James Van Oosting, Dr Molly Smith, and Dr Bert Wachsmuth and the members of the Department of Mathematics and Computer Science We also thank Colonel Gary

W Krahn of the United States Military Academy.

Ms Barbara Gilson and Ms Adrinda Kelly of McGraw-Hill were always ready to provide any needed guidance and Dr Carol Cooper, our contact in the United Kingdom, was equally helpful Thank you, one and all.

Chapter 1 Basic Concepts 1

Differential Equations 1 Notation 2

Solutions 2 Initial-Value and Boundary-Value Problems 2

Mathematical Models 9 The “Modeling Cycle” 9 Qualitative Methods 10

Standard Form and Differential Form 14 Linear Equations 14

Bernoulli Equations 14 Homogeneous Equations 15 Separable Equations 15 Exact Equations 15

General Solution 21 Solutions to the Initial-Value Problem 21 Reduction of Homogeneous Equations 22

Defining Properties 31 Method of Solution 31 Integrating Factors 32

Method of Solution 42 Reduction of Bernoulli Equations 42

CONTENTS

Chapter 7 Applications of First-Order Differential

Growth and Decay Problems 50 Temperature Problems 50 Falling Body Problems 51 Dilution Problems 52 Electrical Circuits 52 Orthogonal Trajectories 53

Linear Differential Equations 73 Linearly Independent Solutions 74 The Wronskian 74

Nonhomogeneous Equations 74

Introductory Remark 83 The Characteristic Equation 83 The General Solution 84

The Characteristic Equation 89 The General Solution 90

Simple Form of the Method 94 Generalizations 95

Modifications 95 Limitations of the Method 95

The Method 103 Scope of the Method 104

Chapter 13 Initial-Value Problems for Linear

Chapter 14 Applications of Second-Order Linear

Spring Problems 114 Electrical Circuit Problems 115 Buoyancy Problems 116 Classifying Solutions 117

Matrices and Vectors 131 Matrix Addition 131 Scalar and Matrix Multiplication 132 Powers of a Square Matrix 132 Differentiation and Integration of Matrices 132 The Characteristic Equation 133

Definition 140

Computation of eAt 140

An Example 148

Reduction of an nthOrder Equation 149 Reduction of a System 150

Qualitative Methods 157 Direction Fields 157 Euler’s Method 158 Stability 158

General Remarks 176 Modified Euler’s Method 177 Runge–Kutta Method 177 Adams–Bashford–Moulton Method 177 Milne’s Method 177

Starting Values 178 Order of a Numerical Method 178

Chapter 20 Numerical Methods for Solving Second-Order

Second-Order Differential Equations 195 Euler’s Method 196

Runge–Kutta Method 196 Adams–Bashford–Moulton Method 196

Definition 211 Properties of Laplace Transforms 211 Functions of Other Independent Variables 212

Definition 224 Manipulating Denominators 224 Manipulating Numerators 225

Convolutions 233 Unit Step Function 233 Translations 234

Laplace Transforms of Derivatives 242 Solutions of Differential Equations 243

The Method 249

Solution of the Initial-Value Problem 254 Solution with No Initial Conditions 255

Second-Order Equations 262 Analytic Functions and Ordinary Points 262 Solutions Around the Origin of Homogeneous Equations 263

Solutions Around the Origin of Nonhomogeneous Equations 263 Initial-Value Problems 264

Solutions Around Other Points 264

Regular Singular Points 275 Method of Frobenius 275 General Solution 276

Classical Differential Equations 290 Polynomial Solutions and Associated Concepts 290

Gamma Function 295 Bessel Functions 295 Algebraic Operations on Infinite Series 296

Introductory Concepts 304 Solutions and Solution Techniques 305

Standard Form 309 Solutions 310 Eigenvalue Problems 310 Sturm–Liouville Problems 310 Properties of Sturm–Liouville Problems 310

Piecewise Smooth Functions 318 Fourier Sine Series 319

Fourier Cosine Series 319

Introduction 325 Classifications 325 Solutions 326

APPENDIX B Some Comments about Technology 336

Introductory Remarks 336 T1-89 337

Differential Equations

t variable If the unknown function depends on two or more independent variables, the

differ-one independent

s a

k will be ordinary differential equations

Example 1.2. Eqquations (1.1) through (1.4) are examples, of ordinary differential equations, since the unknown function y

he variable

depends solely on th x Equation (1.5) is a partial differential equation, since y depends on both the independent

variables t and x.

The order of a a differential equation is the order of the highest derivative appearing in the equation.

Example 1.3. Eqquation (1.1) is a first-order differential equation; (1.2), (1.4), and (1.5) are second-order differential

y t

y x

3 2

2 2

dy dxy

2 2

The expressions y , y , y , y(4), …, y(n)are often used to represent, respectively, the first, second, third, fourth,

…, nth derivatives of y with respect to the independent variable under consideration Thus, y represents d2 2y /dx2

if the independent variable is x, but represents d2 2y /dp2if the independent variable is p Observe that parentheses are used in y(n)to distinguish it from the nth power, y(n) If the independent variable is time, usually denoted by t,

primes are often replaced by dots Thus, ÿ, and represent dy/dt, d2 2y /dt2, and d3y/dt3, respectively.

SOLUTIONS

A solution of a differential equation in the unknown function y and the independent variable x on the interval

Ᏽ, is a function y(x) that satisfies the differential equation identically for all x in Ᏽ.

Example 1.4. Is y(x) = c1sin 2x2 + c2cos 2x22 , where c1and c2are arbitrary constants, a solution of y + 4y = 0?

Thus, y = c1sin 2x2 + c2cos 2x22 satisfies the differential equation for all values of x and is a solution on the interval ( , )

Example 1.5. Determine whether y = x2 1 is a solution of (y)4+ y2= 1

Note that the left side of the differential equation must be nonnegative for every real function y(x) and any x, since itt

is the sum of terms raised to the second and fourth powers, while the right side of the equation is negative Since no function

yy(x) will satisfy this equation, the given differential equation has no solution.

We see that some differential equations have infinitely many solutions (Example 1.4), whereas other ferential equations have no solutions (Example 1.5) It is also possible that a differential equation has exactly

dif-one solution Consider (y )4+ y2= 0, which for reasons identical to those given in Example 1.5 has only one solution y ∫ 0.

A particular solution of a differential equation is any one solution The general solution of a differential

equation is the set of all solutions.

Example 1.6. The general solution to the differential equation in Example 1.4 can be shown to be (see Chapters 8 and 9)

yy = c1sin 2x2 + c2cos 2x2 That is, every particular solution of the differential equation has this general form A few particularr

solutions are: (a) y = 5 sin 2x2  3 cos 2x22 (choose c1= 5 and c2=  3), (b) y = sin 2x22 (choose c1= 1 and c2= 0), and (c) y ∫ 0 (choose c1= c2= 0)

The general solution of a differential equation cannot always be expressed by a single formula As an example

consider the differential equation y + y2= 0, which has two particular solutions y = 1/x // and y ∫ 0.

INITIAL-VALUE AND BOUNDARY-VALUE PROBLEMS

A differential equation along with subsidiary conditions on the unknown function and its derivatives, all

given at the same value of the independent variable, constitutes an initial-value problem The subsidiary tions are initial conditions If the subsidiary conditions are given at more than one value of the independent t

condi-variable, the problem is a boundary-value problem and the conditions are boundary conditions.

y

y,

Example 1.7. The problem y + 2y = e x ; y (

ditions are both given at x= y + 2y = e x ; y(0) = 1, y(1) = 1 is a boundary-value problem, because the two subsidiary conditions are given at the different values x = 0 and x = 1.

A solution to an initial-value or boundary-value problem is a function y(x) that both solves the differential

equation and satisfies all given subsidiary conditions.

(d d) Fourth-order, because the highest-order derivative is the fourth Raising derivatives to various powers does nott

alter the number of derivatives involved The unknown function is b; the independent variable is p.

1.2. Determine the order, unknown function, and the independent variable in each of the following differential equations:

(c) (d) d 17y(4) t6 6 (2)y  4.2y5= 3 cos t

(a) Second-order The unknown function is x; the independent variable is y.

(b) First-order, because the highest-order derivative is the first even though it is raised to the second power The unknown function is x; the independent variable is y.

(c) Third-order The unknown function is x; the independent variable is t.

(d) Fourth-order The unknown function is d y; the independent variable is t Note the difference in notation between the fourth derivative y(4), with parentheses, and the fifth power y5, without parentheses

1.3. Determine whether y(x) = 2ex + xex is a solution of y + 2y + y = 0.

Differentiating y(x), it follows that

y d x

2

2 21

4 4

d b dp

1.4. Is y(x) ∫ 1 a solution of y + 2y + y = x?

From y(x) ∫ 1 it follows that y(x) ∫ 0 and y (x) ∫ 0 Substituting these values into the differential equation,

we obtain

y ... differential equations in Problem 3.7 are Bernoulli equations.

All of the linear equations are Bernoulli equations with n=0 In addition, three of the nonlinearr

equations, ... the general framework of differential equations, the word “homogeneous” has an entirely different t meaning (see Chapter 8) Only in the context of first-order differential equations does “homogeneous”... equation, the given differential equation has no solution.

We see that some differential equations have infinitely many solutions (Example 1.4), whereas other ferential equations have