advanced engineering mathematics with matlab fourth edition pdf

Tallarida Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow... 2.1 Homogeneous Linear Equations with Constant Coefficients 492.4 Method of Undetermined Coefficients

Advances in Applied Mathematics

Advances in Applied Mathematics

Series Editor: Daniel Zwillinger

Published Titles

Advanced Engineering Mathematics with MATLAB, Fourth Edition

Dean G Duffy

CRC Standard Curves and Surfaces with Mathematica®, Third Edition

David H von Seggern

Dynamical Systems for Biological Modeling: An Introduction

Fred Brauer and Christopher Kribs

Fast Solvers for Mesh-Based Computations Maciej Paszy´nski

Green’s Functions with Applications, Second Edition Dean G Duffy Introduction to Financial Mathematics Kevin J Hastings

Linear and Integer Optimization: Theory and Practice, Third Edition

Gerard Sierksma and Yori Zwols

Markov Processes James R Kirkwood

Pocket Book of Integrals and Mathematical Formulas, 5th Edition

Ronald J Tallarida

Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow

Advanced

Engineering Mathematics

FOURTH EDITION

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CLASSIC ENGINEERING MATHEMATICS

2.1 Homogeneous Linear Equations with Constant Coefficients 49

2.4 Method of Undetermined Coefficients 66

3.4 Row Echelon Form and Gaussian Elimination 111

3.5 Eigenvalues and Eigenvectors 124

3.6 Systems of Linear Differential Equations 133

5.4 Fourier Series with Phase Angles 211

5.5 Complex Fourier Series 213

5.6 The Use of Fourier Series in the Solution of Ordinary Differential Equations 217

6.2 Orthogonality of Eigenfunctions 249

6.3 Expansion in Series of Eigenfunctions 251

6.4 A Singular Sturm-Liouville Problem: Legendre’s Equation 256

6.5 Another Singular Sturm-Liouville Problem: Bessel’s Equation 271

0 0.2 0.4 0.6 0.8 1

7.2 Initial Conditions: Cauchy Problem 300

7.3 Separation of Variables 300

7.5 Numerical Solution of the Wave Equation 325

0 0.5 1

0 2

8.1 Derivation of the Heat Equation 337

8.2 Initial and Boundary Conditions 339

8.3 Separation of Variables 340

8.4 Numerical Solution of the Heat Equation 377

−1

0 1

9.2 Boundary Conditions 387

9.3 Separation of Variables 388

9.4 Poisson’s Equation on a Rectangle 425

9.5 Numerical Solution of Laplace’s Equation 428

9.6 Finite Element Solution of Laplace’s Equation 433

10.5 The Cauchy-Goursat Theorem 460

10.6 Cauchy’s Integral Formula 463

10.7 Taylor and Laurent Expansions and Singularities 466

10.9 Evaluation of Real Definite Integrals 477

10.10 Cauchy’s Principal Value Integral 485

11.2 Fourier Transforms Containing the Delta Function 518

11.3 Properties of Fourier Transforms 520

11.4 Inversion of Fourier Transforms 532

11.5 Convolution 544

11.6 The Solution of Ordinary Differential Equations by Fourier Transforms 547

11.7 The Solution of Laplace’s Equation on the Upper Half-Plane 549

11.8 The Solution of the Heat Equation 551

12.1 Definition and Elementary Properties 559

12.2 The Heaviside Step and Dirac Delta Functions 563

12.4 The Laplace Transform of a Periodic Function 579

12.5 Inversion by Partial Fractions: Heaviside’s Expansion Theorem 581

12.8 Solution of Linear Differential Equations with Constant Coefficients 597

12.9 Inversion by Contour Integration 613

12.10 The Solution of the Wave Equation 619

12.11 The Solution of the Heat Equation 637

12.12 The Superposition Integral and the Heat Equation 651

12.13 The Solution of Laplace’s Equation 662

13.1 The Relationship of the Z-Transform to the Laplace Transform 668

13.2 Some Useful Properties 674

13.4 Solution of Difference Equations 691

13.5 Stability of Discrete-Time Systems 697

exact Hilbert transform

computed Hilbert transform

15.1 What Is a Green’s Function? 725

15.2 Ordinary Differential Equations 732

15.3 Joint Transform Method 752

16.4 Continuous Random Variables 822

16.6 Some Commonly Used Distributions 834

17.3 Two-State Markov Chains 867

17.4 Birth and Death Processes 874

18.1 Random Differential Equations 896

18.2 Random Walk and Brownian Motion 905

18.3 Itˆo’s Stochastic Integral 916

18.5 Stochastic Differential Equations 928

18.6 Numerical Solution of Stochastic Differential Equations 936

Answers to the Odd-Numbered Problems 945

MATLABis a registered trademark of

The MathWorks Inc

24 Prime Park WayNatick, MA 01760-1500Phone: (508) 647-7000Email: info@mathworks.comwww.mathworks.com

Dean G Duffy received his bachelor of science in geophysics from Case Institute ofTechnology (Cleveland, Ohio) and his doctorate of science in meteorology from the Mas-sachusetts Institute of Technology (Cambridge, Massachusetts) He served in the UnitedStates Air Force from September 1975 to December 1979 as a numerical weather predictionofficer After his military service, he began a twenty-five-year (1980 to 2005) associationwith NASA at the Goddard Space Flight Center (Greenbelt, Maryland) where he focused onnumerical weather prediction, oceanic wave modeling, and dynamical meteorology He alsowrote papers in the areas of Laplace transforms, antenna theory, railroad tracks, and heatconduction In addition to his NASA duties, he taught engineering mathematics, differentialequations, and calculus at the United States Naval Academy (Annapolis, Maryland) andthe United States Military Academy (West Point, New York) Drawing from his teachingexperience, he has written several books on transform methods, engineering mathematics,Green’s functions, and mixed-boundary value problems

Today’s STEM (science, technology, engineering, and mathematics) student must ter vast quantities of applied mathematics This is why I wrote Advanced EngineeringMathematics with MATLAB Three assumptions underlie its structure: (1) All studentsneed a firm grasp of the traditional disciplines of ordinary and partial differential equa-tions, vector calculus, and linear algebra (2) The digital revolution will continue Thusthe modern student must have a strong foundation in transform methods because theyprovide the mathematical basis for electrical and communication studies (3) The biologi-cal revolution will become more mathematical and require an understanding of stochastic(random) processes Already, stochastic processes play an important role in finance, thephysical sciences, and engineering These techniques will enjoy an explosive growth in thebiological sciences For these reasons, an alternative title for this book could be AdvancedEngineering Mathematics for the Twenty-First Century

mas-This is my fourth attempt at realizing these goals It continues the tradition of includingtechnology into the conventional topics of engineering mathematics Of course, I took thisopportunity to correct misprints and include new examples, problems, and projects I nowuse the small rectangle⊓⊔ to separate the end of an example or theorem from the continuingtext The two major changes are a section on conformal mapping (Section 10.11) and anew chapter on stochastic calculus

A major change is the reorganization of the order of the chapters In line with mygoals I have subdivided the material into three groups: classic engineering mathematics,transform methods, and stochastic processes In its broadest form, there are two generaltracks:

Differential Equations Course: Most courses on differential equations cover three eral topics: fundamental techniques and concepts, Laplace transforms, and separation ofvariable solutions to partial differential equations

gen-The course begins with first- and higher-order ordinary differential equations,Chapters

1and2, respectively After some introductory remarks,Chapter 1devotes itself to ing general methods for solving first-order ordinary differential equations These methods

present-include separation of variables, employing the properties of homogeneous, linear, and exactdifferential equations, and finding and using integrating factors.

The reason most students study ordinary differential equations is for their use in mentary physics, chemistry, and engineering courses Because these differential equationscontain constant coefficients, we focus on how to solve them in Chapter 2, along with

ele-a detele-ailed ele-anele-alysis of the simple, dele-amped, ele-and forced hele-armonic oscillele-ator Furthermore,

we include the commonly employed techniques of undetermined coefficients and variation

of parameters for finding particular solutions Finally, the special equation of Euler andCauchy is included because of its use in solving partial differential equations in sphericalcoordinates

Some courses include techniques for solving systems of linear differential equations Achapter on linear algebra (Chapter 3) is included if that is a course objective

After these introductory chapters, the course would next turn to Laplace transforms.Laplace transforms are useful in solving nonhomogeneous differential equations where theinitial conditions have been specified and the forcing function “turns on and off.” Thegeneral properties are explored inSection 12.1toSection 12.7; the actual solution technique

is presented inSection 12.8

Most differential equations courses conclude with a taste of partial differential tions via the method of separation of variables This topic usually begins with a quickintroduction to Fourier series, Sections 5.1 to 5.4, followed by separation of variables as

equa-it applies to the heat (Sections 8.1–8.3), wave (Sections 7.1–7.3), or Laplace’s equations(Sections 9.1–9.3) The exact equation that is studied depends upon the future needs of thestudents

Engineering Mathematics Course: This book can be used in a wide variety of neering mathematics classes In all cases the student should have seen most of the material

engi-inChapters 1and2 There are at least four possible combinations:

•Option A: The course is a continuation of a calculus reform sequence where elementarydifferential equations have been taught This course begins with Laplace transforms andseparation of variables techniques for the heat, wave, and/or Laplace’s equations, as outlinedabove The course then concludes with either vector calculus or linear algebra Vectorcalculus is presented in Chapter 4 and focuses on the gradient operator as it applies toline integrals, surface integrals, the divergence theorem, and Stokes’ theorem Chapter

3 presents linear algebra as a method for solving systems of linear equations and includessuch topics as matrices, determinants, Cramer’s rule, and the solution of systems of ordinarydifferential equations via the classic eigenvalue problem

•Option B: This is the traditional situation where the student has already studied tial equations in another course before he takes engineering mathematics Here separation ofvariables is retaught from the general viewpoint of eigenfunction expansions Sections 9.1–

differen-9.3explain how any piece-wise continuous function can be reexpressed in an eigenfunctionexpansion using eigenfunctions from the classic Sturm-Liouville problem Furthermore, weinclude two sections that focus on Bessel functions (Section 6.5) and Legendre polynomials(Section 6.4) These eigenfunctions appear in the solution of partial differential equations

in cylindrical and spherical coordinates, respectively

The course then covers linear algebra and vector calculus as given in Option A

•Option C : I originally wrote this book for an engineering mathematics course given tosophomore and junior communication, systems, and electrical engineering majors at theU.S Naval Academy In this case, you would teach all of Chapter 10 with the possible

exception ofSection 10.10on Cauchy principal-value integrals This material was added toprepare the student for Hilbert transforms,Chapter 14.

Because most students come to this course with a good knowledge of differential tions, we begin with Fourier series,Chapter 5, and proceed throughChapter 14 Chapter 11

equa-generalizes the Fourier series to aperiodic functions and introduces the Fourier transform.This leads naturally to Laplace transforms,Chapter 12 Throughout these chapters, I makeuse of complex variables in the treatment and inversion of the transforms

With the rise of digital technology and its associated difference equations, a version

of the Laplace transform, the z-transform, was developed Chapter 13 introduces the transform by first giving its definition and then developing some of its general properties

z-We also illustrate how to compute the inverse by long division, partial fractions, and tour integration Finally, we use z-transforms to solve difference equations, especially withrespect to the stability of the system

con-Finally, there is a chapter on the Hilbert transform With the explosion of interest incommunications, today’s engineer must have a command of this transform The Hilberttransform is introduced in Section 14.1 and its properties are explored in Section 14.2.Two important applications of Hilbert transforms are introduced inSections 14.3and14.4,namely the concept of analytic signals and the Kramers-Kronig relationship

•Option D: Many engineering majors now require a course in probability and statisticsbecause of the increasing use of probabilistic concepts in engineering analysis To incorpo-rate this development into an engineering mathematics course we adopt a curriculum thatbegins with Fourier transforms (minus inversion by complex variables) given in Chapter

11 The remaining portion involves the fundamental concepts of probability presented in

Chapter 16 and random processes in Chapter 17 Chapter 16 introduces the student tothe concepts of probability distributions, mean, and variance because these topics appear

so frequently in random processes Chapter 17explores common random processes such asPoisson processes and birth and death Of course, this course assumes a prior knowledge

of ordinary differential equations and Fourier series

A unique aspect of this book appears in Chapter 18, which is devoted to stochasticcalculus We start by exploring deterministic differential equations with a stochastic forcing.Next, the important stochastic process of Brownian motion is developed in depth Using thisBrownian motion, we introduce the concept of (Itˆo) stochastic integration, Itˆo’s lemma, andstochastic differential equations The chapter concludes with various numerical methods tointegrate stochastic differential equations

In addition to the revisions of the text and topics covered in this new addition, MATLAB

is still employed to reinforce the concepts that are taught Of course, this book still continues

my principle of including a wealth of examples from the scientific and engineering literature.The answers to the odd problems are given in the back of the book, while worked solutions

to all of the problems are available from the publisher Most of the MATLABscripts may

be found at http://www.crcpress.com/product/isbn/9781439816240

Z x 0

e−y2dyΓ(x) gamma function

In(x) modified Bessel function of the first kind and order n

Jn(x) Bessel function of the first kind and order n

Kn(x) modified Bessel function of the second kind and order n

Pn(x) Legendre polynomial of order n

ℜ(z) real part of the complex variable z

A differential equation is any equation that contains the derivatives or differentials ofone or more dependent variables with respect to one or more independent variables Becausemany of the known physical laws are expressed as differential equations, a sound knowledge

of how to solve them is essential In the next two chapters we present the fundamentalmethods for solving ordinary differential equations - a differential equation that containsonly ordinary derivatives of one or more dependent variables Later, in Sections 11.6 and

12.8, we show how transform methods can be used to solve ordinary differential equations,while systems of linear ordinary differential equations are treated inSection 3.6 Solutionsfor partial differential equations—a differential equation involving partial derivatives of one

or more dependent variables of two or more independent variables—are given in Chapters

7, 8, and9

1.1 CLASSIFICATION OF DIFFERENTIAL EQUATIONS

Differential equations are classified three ways: by type, order , and linearity Thereare two types: ordinary and partial differential equations, which have already been defined.Examples of ordinary differential equations include

dy

dx− 2y = x, (1.1.1)(x− y) dx + 4y dy = 0, (1.1.2)du

dx+dv

dx = 1 + 5x, (1.1.3)

differ-The order of a differential equation is given by the highest-order derivative For ple,

2

− y = sin(x) (1.1.8)

is a third-order ordinary differential equation Because we can rewrite

(x + y) dy− x dx = 0 (1.1.9)as

of linear first-, second-, and third-order ordinary differential equations are

(x + 1) dy− y dx = 0, (1.1.14)

y′′+ 3y′+ 2y = ex, (1.1.15)and

xd

3y

dx3 − (x2+ 1)dy

dx + y = sin(x), (1.1.16)respectively If the differential equation is not linear, then it is nonlinear Examples ofnonlinear first-, second-, and third-order ordinary differential equations are

5

+ 2xy = sin(x), (1.1.18)and

yy′′′+ 2y = ex, (1.1.19)respectively

At this point it is useful to highlight certain properties that all differential equationshave in common regardless of their type, order, and whether they are linear or not First, it

is not obvious that just because we can write down a differential equation, a solution exists.The existence of a solution to a class of differential equations constitutes an important aspect

of the theory of differential equations Because we are interested in differential equationsthat arise from applications, their solution should exist In Section 1.2 we address thisquestion further

Quite often a differential equation has the solution y = 0, a trivial solution Forexample, if f (x) = 0 in Equation 1.1.13, a quick check shows that y = 0 is a solution.Trivial solutions are generally of little value

Another important question is how many solutions does a differential equation have?

In physical applications uniqueness is not important because, if we are lucky enough toactually find a solution, then its ties to a physical problem usually suggest uniqueness.Nevertheless, the question of uniqueness is of considerable importance in the theory ofdifferential equations Uniqueness should not be confused with the fact that many solutions

to ordinary differential equations contain arbitrary constants, much as indefinite integrals

in integral calculus A solution to a differential equation that has no arbitrary constants iscalled a particular solution

• Example 1.1.1

Consider the differential equation

dy

dx = x + 1, y(1) = 2. (1.1.20)This condition y(1) = 2 is called an initial condition and the differential equation plus theinitial condition constitute an initial-value problem Straightforward integration yields

y(x) =

Z(x + 1) dx + C = 12x2+ x + C (1.1.21)Equation 1.1.21 is the general solution to the differential equation, Equation 1.1.20, because

it is a solution to the differential equation for every choice of C However, if we now satisfy

the initial condition y(1) = 2, we obtain a particular solution This is done by substitutingthe corresponding values of x and y into Equation 1.1.21, or

2 = 12(1)2+ 1 + C = 32+ C, or C =12 (1.1.22)Therefore, the solution to the initial-value problem Equation 1.1.20 is the particular solution

y(x) = (x + 1)2/2 (1.1.23)

⊓Finally, it must be admitted that most differential equations encountered in the “real”world cannot be written down either explicitly or implicitly For example, the simple differ-ential equation y′= f (x) does not have an analytic solution unless you can integrate f (x).This begs the question of why it is useful to learn analytic techniques for solving differentialequations that often fail us The answer lies in the fact that differential equations that wecan solve share many of the same properties and characteristics of differential equationswhich we can only solve numerically Therefore, by working with and examining the dif-ferential equations that we can solve exactly, we develop our intuition and understandingabout those that we can only solve numerically

ProblemsFind the order and state whether the following ordinary differential equations are linear ornonlinear:

By multiplying both sides of Equation 1.2.1 by dx, we obtain

For this technique to work, we must be able to rewrite the differential equation so that all

of the y dependence appears on one side of the equation while the x dependence is on theother Finally we must be able to carry out the integration on both sides of the equation.One of the interesting aspects of our analysis is the appearance of the arbitrary constant

C in Equation 1.2.3 To evaluate this constant we need more information The mostcommon method is to require that the dependent variable give a particular value for aparticular value of x Because the independent variable x often denotes time, this condition

is usually called an initial condition, even in cases when the independent variable is nottime

its solution is simply

−ye−y− e−y = ln|x| + C (1.2.6)

subject to the initial condition y(0) = 1

Multiplying Equation 1.2.7 by dx, we find that

dy + y dx = xexy dx, (1.2.8)or

dy

y = (xe

x

− 1) dx (1.2.9)

A quick check shows that the left side of Equation 1.2.9 contains only the dependent variable

y while the right side depends solely on x and we have separated the variables onto one side

or the other Finally, integrating both sides of this equation, we have

ln(y) = xex− ex− x + C (1.2.10)Since y(0) = 1, C = 1 and

y(x) = exp[(x− 1)ex+ 1− x] (1.2.11)

In addition to the tried-and-true method of solving ordinary differential equations byhand, scientific computational packages such as MATLABprovide symbolic toolboxes thatare designed to do the work for you In the present case, typing

dsolve(’Dy+y=x*exp(x)*y’,’y(0)=1’,’x’)

yields

ans =

1/exp(-1)*exp(-x+x*exp(x)-exp(x))

which is equivalent to Equation 1.2.11

Our success here should not be overly generalized Sometimes these toolboxes givethe answer in a rather obscure form or they fail completely For example, in the previousexample, MATLABgives the answer

−dyy2 =dx

x2, or 1

y =−x1 + C, or y = x

Cx− 1. (1.2.13)Equation 1.2.13 shows the wide variety of solutions possible for an ordinary differentialequation For example, if we require that y(0) = 0, then there are infinitely many differentsolutions satisfying this initial condition because C can take on any value On the otherhand, if we require that y(0) = 1, there is no solution because we cannot choose any constant

C such that y(0) = 1 Finally, if we have the initial condition that y(1) = 2, then there isonly one possible solution corresponding to C = 32

Consider now the trial solution y = 0 Does it satisfy Equation 1.2.12? Yes, it does

On the other hand, there is no choice of C that yields this solution The solution y = 0 iscalled a singular solution to this equation Singular solutions are solutions to a differentialequation that cannot be obtained from a solution with arbitrary constants

c = 0

0.5 1

x

c = 4

Figure 1.2.1: The solution to Equation 1.2.13 when C = −2, 0, 2, 4.

Finally, we illustrate Equation 1.2.13 using MATLAB This is one of MATLAB’s strengths

— the ability to convert an abstract equation into a concrete picture Here the MATLAB

yieldsFigure 1.2.1, which illustrates Equation 1.2.13 when C =−2, 0, 2, and 4 ⊓

The previous example showed that first-order ordinary differential equations may have aunique solution, no solution, or many solutions From a complete study2of these equations,

we have the following theorem:

2 The proof of the existence and uniqueness of first-order ordinary differential equations is beyond the scope of this book See Ince, E L., 1956: Ordinary Differential Equations Dover Publications, Inc.,

Theorem: Existence and Uniqueness

Suppose some real-valued function f (x, y) is continuous on some rectangle in the plane containing the point (a, b) in its interior Then the initial-value problem

xy-dy

dx = f (x, y), y(a) = b, (1.2.14)has at least one solution on the same open interval I containing the point x = a Further-more, if the partial derivative ∂f /∂y is continuous on that rectangle, then the solution isunique on some (perhaps smaller) open interval I0 containing the point x = a ⊓

• Example 1.2.4

Consider the initial-value problem y′= 3y1/3/2 with y(0) = 1 Here f (x, y) = 3y1/3/2and fy = y−2/3/2 Because fy is continuous over a small rectangle containing the point(0, 1), there is a unique solution around x = 0, namely y = (x + 1)3/2, which satisfies thedifferential equation and the initial condition On the other hand, if the initial conditionreads y(0) = 0, then fy is not continuous on any rectangle containing the point (0, 0) andthere is no unique solution For example, two solutions to this initial-value problem, valid

on any open interval that includes x = 0, are y1(x) = x3/2and

y2(x) =

(x− 1)3/2, x≥ 1,

0, x < 1 (1.2.15)

⊓

• Example 1.2.5: Hydrostatic equation

Consider an atmosphere where its density varies only in the vertical direction Thepressure at the surface equals the weight per unit horizontal area of all of the air from sealevel to outer space As you move upward, the amount of air remaining above decreasesand so does the pressure This is why we experience pressure sensations in our ears whenascending or descending in an elevator or airplane If we rise the small distance dz, theremust be a corresponding small decrease in the pressure, dp This pressure drop must equalthe loss of weight in the column per unit area,−ρg dz Therefore, the pressure is governed

by the differential equation

dp =−ρg dz, (1.2.16)commonly called the hydrostatic equation

To solve Equation 1.2.16, we must express ρ in terms of pressure For example, in

an isothermal atmosphere at constant temperature Ts, the ideal gas law gives p = ρRTs,where R is the gas constant Substituting this relationship into our differential equationand separating variables yields

Thus, the pressure (and density) of an isothermal atmosphere decreases exponentially withheight In particular, it decreases by e−1 over the distance RTs/g, the so-called “scale

• Example 1.2.6: Terminal velocity

As an object moves through a fluid, its viscosity resists the motion Let us find themotion of a mass m as it falls toward the earth under the force of gravity when the dragvaries as the square of the velocity

From Newton’s second law, the equation of motion is

mdv

dt = mg− CDv2, (1.2.19)where v denotes the velocity, g is the gravitational acceleration, and CD is the drag coeffi-cient We choose the coordinate system so that a downward velocity is positive

Equation 1.2.19 can be solved using the technique of separation of variables if we changefrom time t as the independent variable to the distance traveled x from the point of release.This modification yields the differential equation

mvdv

dx = mg− CDv2, (1.2.20)since v = dx/dt Separating the variables leads to

v dv

1− kv2/g = g dx, (1.2.21)or

v2(x) = g

k 1− e−2kx
(1.2.23)Thus, as the distance that the object falls increases, so does the velocity, and it eventuallyapproaches a constant valuep

g/k, commonly known as the terminal velocity

Because the drag coefficient CD varies with the superficial area of the object whilethe mass depends on the volume, k increases as an object becomes smaller, resulting in asmaller terminal velocity Consequently, although a human being of normal size will acquire

a terminal velocity of approximately 120 mph, a mouse, on the other hand, can fall any

• Example 1.2.7: Interest rate

Consider a bank account that has been set up to pay out a constant rate of P dollarsper year for the purchase of a car This account has the special feature that it pays anannual interest rate of r on the current balance We would like to know the balance in theaccount at any time t

Although financial transactions occur at regularly spaced intervals, an excellent proximation can be obtained by treating the amount in the account x(t) as a continuousfunction of time governed by the equation

ap-x(t + ∆t)≈ x(t) + rx(t)∆t − P ∆t, (1.2.24)where we have assumed that both the payment and interest are paid in time increments

of ∆t As the time between payments tends to zero, we obtain the first-order ordinarydifferential equation

of money in the account will grow without bound Finally, the case x(0) = P/r is theequilibrium case where the amount of money paid out balances the growth of money due

to interest so that the account always has the balance of P/r ⊓

• Example 1.2.8: Steady-state flow of heat

When the inner and outer walls of a body, for example the inner and outer walls

of a house, are maintained at different constant temperatures, heat will flow from thewarmer wall to the colder one When each surface parallel to a wall has attained a constanttemperature, the flow of heat has reached a steady state In a steady-state flow of heat,each surface parallel to a wall, because its temperature is now constant, is referred to as anisothermal surface Isothermal surfaces at different distances from an interior wall will havedifferent temperatures In many cases the temperature of an isothermal surface is only afunction of its distance x from the interior wall, and the rate of flow of heat Q in a unittime across such a surface is proportional both to the area A of the surface and to dT /dx,where T is the temperature of the isothermal surface Hence,

Q =−κAdTdx, (1.2.27)where κ is called the thermal conductivity of the material between the walls

In place of a flat wall, let us consider a hollow cylinder whose inner and outer surfacesare located at r = r1 and r = r2, respectively At steady state, Equation 1.2.27 becomes

Qr=−κAdTdr =−κ(2πrL)dTdr, (1.2.28)assuming no heat generation within the cylindrical wall

We can find the temperature distribution inside the cylinder by solving Equation 1.2.28along with the appropriate conditions on T (r) at r = r1 and r = r2 (the boundary con-ditions) To illustrate the wide choice of possible boundary conditions, let us require thatinner surface is maintained at the temperature T1 We assume that along the outer surface

heat is lost by convection to the environment, which has the temperature T∞ This heatloss is usually modeled by the equation

κ dTdr

r=r 2

=−h(T − T∞), (1.2.29)where h > 0 is the convective heat transfer coefficient Upon integrating Equation 1.2.28,

T (r) =−2πκLQr ln(r) + C, (1.2.30)where Qr is also an unknown Substituting Equation 1.2.30 into the boundary conditions,

we obtain

T (r) = T1+ Qr

2πκLln(r1/r), (1.2.31)with

Qr= 2πκL(T1− T∞)κ/r2+ h ln(r2/r1). (1.2.32)

As r2 increases, the first term in the denominator of Equation 1.2.32 decreases while thesecond term increases Therefore, Qr has its largest magnitude when the denominator issmallest, assuming a fixed numerator This occurs at the critical radius rcr= κ/h, where

Qmaxr = 2πκL(T1− T∞)

1 + ln(rcr/r1) . (1.2.33)

⊓

• Example 1.2.9: Population dynamics

Consider a population P (t) that can change only by a birth or death but not by gration or emigration If B(t) and D(t) denote the number of births or deaths, respectively,

immi-as a function of time t, the birth rate and death rate (in births or deaths per unit time) is

dB

dt, (1.2.34)and

dD

dt . (1.2.35)Now,

P′(t) = [b(t)− d(t)]P (t) (1.2.39)

When the birth and death rates are constants, namely b and d, respectively, the ulation evolves according to

pop-P (t) = pop-P (0) exp

b− d
t

⊓

• Example 1.2.10: Logistic equation

The study of population dynamics yields an important class of first-order, nonlinear,ordinary differential equations: the logistic equation This equation arose in Pierre Fran¸coisVerhulst’s (1804–1849) study of animal populations.3 If x(t) denotes the number of species

in the population and k is the (constant) environment capacity (the number of species thatcan simultaneously live in the geographical region), then the logistic or Verhulst’s equationis

x′= ax(k− x)/k, (1.2.41)where a is the population growth rate for a small number of species

To solve Equation 1.2.41, we rewrite it as

dx(1− x/k)x=

dx

x +

x/k

1− x/kdx = r dt. (1.2.42)Integration yields

ln|x| − ln |1 − x/k| = rt + ln(C), (1.2.43)or

As t→ ∞, x(t) → k, the asymptotically stable solution ⊓

• Example 1.2.11: Chemical reactions

Chemical reactions are often governed by first-order ordinary differential equations.For example, first-order reactions, which describe reactions of the form A → B, yield thekdifferential equation

−a1d[A]dt = k[A], (1.2.46)where k is the rate at which the reaction is taking place Because for every molecule of Athat disappears one molecule of B is produced, a = 1 and Equation 1.2.46 becomes

−d[A]dt = k[A] (1.2.47)

3 Verhulst, P F., 1838: Notice sur la loi que la population suit dans son accroissement Correspond Math Phys., 10, 113–121.

Integration of Equation 1.2.47 leads to

[A] = [A]0e−kt (1.2.50)The exponential form of the solution suggests that there is a time constant τ , which is calledthe decay time of the reaction This quantity gives the time required for the concentration

of decrease by 1/e of its initial value [A]0 It is given by τ = 1/k

Turning to second-order reactions, there are two cases The first is a reaction betweentwo identical species: A + A→ products The rate expression here isk

−12d[A]dt = k[A]2 (1.2.51)The second case is an overall second-order reaction between two unlike species, given by A+ B→ X In this case, the reaction is first order in each of the reactants A and B and thekrate expression is

−d[A]dt = k[A][B] (1.2.52)Turning to Equation 1.2.51 first, we have by separation of variables

dτ, (1.2.53)

or

1[A] =

1[A]0 + 2kt. (1.2.54)Therefore, a plot of the inverse of A versus time will yield a straight line with slope equal

to 2k and intercept 1/[A]0

With regard to Equation 1.2.52, because an increase in X must be at the expense

of A and B, it is useful to express the rate equation in terms of the concentration of X,[X] = [A]0− [A] = [B]0− [B], where [A]0and [B]0are the initial concentrations Then, thisequation becomes

d[X]

dt = k ([A]0− [X]) ([B]0− [X]) (1.2.55)Separation of variables leads to

Z [X]

[X]0

dξ([A]0− ξ) ([B]0− ξ) = k

Z t 0

Carrying out the integration,

1[A]0− [B]0

10 Setting u = y− x, solve the first-order ordinary differential equation

12 Using the hydrostatic equation, show that the pressure within an atmosphere with thetemperature distribution

T (z) =



T0− Γz, 0≤ z ≤ H,

T0− ΓH, H ≤ z,

13 The voltage V as a function of time t within an electrical circuit4 consisting of acapacitor with capacitance C and a diode in series is governed by the first-order ordinarydifferential equation

14 A glow plug is an electrical element inside a reaction chamber, which either ignites thenearby fuel or warms the air in the chamber so that the ignition will occur more quickly

An accurate prediction of the wire’s temperature is important in the design of the chamber.Assuming that heat convection and conduction are not important,5the temperature T

of the wire is governed by

AdT

dt + B(T

4

− T4) = P,where A equals the specific heat of the wire times its mass, B equals the product of theemissivity of the surrounding fluid times the wire’s surface area times the Stefan-Boltzmannconstant, Ta is the temperature of the surrounding fluid, and P is the power input Thetemperature increases due to electrical resistance and is reduced by radiation to the sur-rounding fluid

Show that the temperature is given by

4Bγ3t

A = 2

tan−1

Tγ

,where γ4= P/B + T4 and T0 is the initial temperature of the wire

15 Let us denote the number of tumor cells by N (t) Then a widely used deterministictumor growth law6 is

dN

dt = bN ln(K/N ),where K is the largest tumor size and 1/b is the length of time required for the specificgrowth to decrease by 1/e If the initial value of N (t) is N (0), find N (t) at any subsequenttime t

4 See Aiken, C B., 1938: Theory of the diode voltmeter Proc IRE , 26, 859–876.

5 See Clark, S K., 1956: Heat-up time of wire glow plugs Jet Propulsion, 26, 278–279.

6 See Hanson, F B., and C Tier, 1982: A stochastic model of tumor growth Math Biosci., 61, 73–100.

to 2k and intercept 1/[A]0

With regard to Equation 1.2.52, because an increase in X must... differential equation

12 Using the hydrostatic equation, show that the pressure within an atmosphere with thetemperature distribution

T (z) =



T0− Γz,... class="page_container" data-page="40">

13 The voltage V as a function of time t within an electrical circuit4 consisting of acapacitor with capacitance C and a diode in series is governed by the first-order