1 .1 .1 General and Particular Solutions A first-order differential equation is any equation involving a first derivative, but no highe r derivative.. Such a solution is called th e gene
Trang 3Advanced Engineering Mathematics, International Student Editio n
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Trang 4PART 1 Ordinary Differential Equations 1
Chapter 1 First-Order Differential Equations 3
1 1 Preliminary Concepts 3
1 1 1 General and Particular Solutions 3
1 1 2 Implicitly Defined Solutions 4
1 1 3 Integral Curves 5
1 1 4 The Initial Value Problem 6
1 1 5 Direction Fields 7
1 2 Separable Equations 1 1
1 2 1 Some Applications of Separable Differential Equations 1 4
1 3 Linear Differential Equations 2 2
1 4 Exact Differential Equations 2 6
1 5 Integrating Factors 3 3
1 5 1 Separable Equations and Integrating Factors 3 7
1 5 2 Linear Equations and Integrating Factors 3 7
1 6 Homogeneous, Bernoulli, and Riccati Equations 3 8
1 6 1 Homogeneous Differential Equations 3 8
1 6 2 The Bernoulli Equation 4 2
1 6 3 The Riccati Equation 4 3
1 7 Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories 46
1 7 1 Mechanics 4 6
1 7 2 Electrical Circuits 5 1
1 7 3 Orthogonal Trajectories 5 3
1 8 Existence and Uniqueness for Solutions of Initial Value Problems 5 8
Chapter 2 Second-Order Differential Equations 6 1
2.1 Preliminary Concepts 6 12.2 Theory of Solutions of y" +p(x)y' + q(x)y = f(x) 6 2
2 2.1 The Homogeneous Equation y" +p(x)y' + q(x) = 0 64
2 2.2 The Nonhomogeneous Equation y" +p(x)y' + q(x)y = f(x) 6 8
2 3 Reduction of Order 69
2 4 The Constant Coefficient Homogeneous Linear Equation 7 3
2 4.1 Case 1 : A2 - 4B > 0 7 3
2 4.2 Case 2 : Az - 4B = 0 74
Trang 52 4 3 Case3 : A2 - 4B<0 7 4
2 4 4 An Alternative General Solution in the Complex Root Case 7 5
2 5 Euler' s Equation 7 8
2 6 The Nonhomogeneous Equation y" +p(x)y' + q(x)y = f(x) 82
2 6 1 The Method of Variation of Parameters 8 2
2 6 2 The Method of Undetermined Coefficients 8 5
2 6 3 The Principle of Superposition 9 1
2 6 4 Higher-Order Differential Equations 9 1
2 7 Application of Second-Order Differential Equations to a Mechanical System 9 3
2 7 1 Unforced Motion 9 5
2 7 2 Forced Motion 98
2 7 3 Resonance 10 0
2 7 4 Beats 10 2
2 7.5 Analogy with an Electrical Circuit 10 3
Chapter 3 The Laplace Transform 10 7
3 1 Definition and Basic Properties 10 7
3 2 Solution of Initial Value Problems Using the Laplace Transform 11 6
3 3 Shifting Theorems and the Heaviside Function 120
3 3 1 The First Shifting Theorem 120
3 3 2 The Heaviside Function and Pulses 12 2
3 3 3 The Second Shifting Theorem 125
3 3 4 Analysis of Electrical Circuits 12 9
3 4 Convolution 13 4
3 5 Unit Impulses and the Dirac Delta Function 13 9
3 6 Laplace Transform Solution of Systems 144
3 7 Differential Equations with Polynomial Coefficients 150
Chapter 4 Series Solutions 155
4.1 Power Series Solutions of Initial Value Problems 15 6
4 2 Power Series Solutions Using Recurrence Relations 16 1
4 3 Singular Points and the Method of Frobenius 16 6
4 4 Second Solutions and Logarithm Factors 17 3
Chapter 5 Numerical Approximation of Solutions 18 1
5 1 Euler's Method 18 2
5 1 1 A Problem in Radioactive Waste Disposal 187
5 2 One-Step Methods 190
5 2 1 The Second-Order Taylor Method 19 0
5 2 2 The Modified Euler Method 19 3
Trang 6Chapter 6 Vectors and Vector Spaces 203
6 1 The Algebra and Geometry of Vectors 203
6 2 The Dot Product 21 1
6 3 The Cross Product 21 7
6 4 The Vector Space R" 223
6 5 Linear Independence, Spanning Sets, and Dimension in R" 22 8
Chapter 7 Matrices and Systems of Linear Equations 23 7
7 1 Matrices 23 8
7 1 1 Matrix Algebra 23 9
7 1 2 Matrix Notation for Systems of Linear Equations 242
7 1 3 Some Special Matrices 243
24 6
7 1 4 Another Rationale for the Definition of Matrix Multiplicatio n
7 1 5 Random Walks in Crystals 24 7
7 2 Elementary Row Operations and Elementary Matrices 25 1
7 3 The Row Echelon Form of a Matrix 25 8
7 4 The Row and Column Spaces of a Matrix and Rank of a Matrix 26 6
7 5 Solution of Homogeneous Systems of Linear Equations 27 2
7 6 The Solution Space of AX = 0 28 0
7 7 Nonhomogeneous Systems of Linear Equations 28 3
7 7 1 The Structure of Solutions of AX = B 28 4
7 7 2 Existence and Uniqueness of Solutions of AX = B 28 5
7 8 Matrix Inverses 293
7 8 1 A Method for Finding A- 1 295
Chapter 8 Determinants 299
8.1 Permutations 29 98.2 Definition of the Determinant 30 18.3 Properties of Determinants 30 3
8 4 Evaluation of Determinants by Elementary Row and Column Operations 30 78.5 Cofactor Expansions 31 1
8 6 Determinants of Triangular Matrices 31 4
8 7 A Determinant Formula for a Matrix Inverse 31 5
8 8 Cramer' s Rule 31 8
8 9 The Matrix Tree Theorem 320
Chapter 9 Eigenvalues, Diagonalization, and Special Matrices 32 3
9 1 Eigenvalues and Eigenvectors 324
9 1 1 Gerschgorin's Theorem 328 '
9 2 Diagonalization of Matrices 33 0
9 3 Orthogonal and Symmetric Matrices 33 9
Trang 79 4 Quadratic Forms 347
9 5 Unitary, Hermitian, and Skew Hermitian Matrices 35 2
Chapter 10 Systems of Linear Differential Equations 36 1
10 1 Theory of Systems of Linear First-Order Differential Equations 36 1
10 1 1 Theory of the Homogeneous System X' = AX 365
10 1 2 General Solution of the Nonhomogeneous System X' = AX + G 37 2
10 2 Solution of X' = AX when A is Constant 374
10 2.1 Solution of X' = AX when A has Complex Eigenvalues 37 7
10 2.2 Solution of X' = AX when A does not have n Linearly Independent
Chapter 11 Qualitative Methods and Systems of Nonlinear Differential Equations 40 3
11 1 Nonlinear Systems and Existence of Solutions 40 3
11 2 The Phase Plane, Phase Portraits and Direction Fields 40 6
11 3 Phase Portraits of Linear Systems 41 3
11 4 Critical Points and Stability 42 4
11 5 Almost Linear Systems 43 1
11 6 Lyapunov ' s Stability Criteria 45 1
11 7 Limit Cycles and Periodic Solutions 46 1
Chapter 12 Vector Differential Calculus 47 5
12 1 Vector Functions of One Variable 47 5
12 2 Velocity, Acceleration, Curvature and Torsion 48 1
12 2 1 Tangential and Normal Components of Acceleration 48 8
12 2 2 Curvature as a Function of t 49 1
12 2 3 The Frenet Formulas 492
12 3 Vector Fields and Streamlines 49 3
12 4 The Gradient Field and Directional Derivatives 49 9
12 4 1 Level Surfaces, Tangent Planes and Normal Lines 50 3
12 5 Divergence and Curl 51 0
12 5 1 A Physical Interpretation of Divergence 51 2
12 5 2 A Physical Interpretation of Curl 513
Trang 813 3 1 A More,Critical Look at Theorem 13 5 53 9
13 4 Surfaces in 3-Space and Surface Integrals 545
13 4.1 Normal Vector to a Surface 54 8
13 4 2 The Tangent Plane to a Surface 55 1
13 4 3 Smooth and Piecewise Smooth Surfaces 55 2
13 4 4 Surface Integrals 55 3
13 5 Applications of Surface Integrals 55 7
13 5 1 Surface Area 557
13 5 2 Mass and Center of Mass of a Shell 55 7
13 5 3 Flux of a Vector Field Across a Surface 56 0
13 6 Preparation for the Integral Theorems of Gauss and Stokes 56 2
13 7 The Divergence Theorem of Gauss 564
13 7 1 Archimedes 's Principle 567
13 7 2 The Heat Equation 568
13 7 3 The Divergence Theorem as aConservation of Mass Principle 570
13 8 The Integral Theorem of Stokes 572
13 8 1 An Interpretation of Curl 57 6
13 8 2 Potential Theory in 3-Space 57 6
PART
Chapter 14 Fourier Series 583
14.1 Why Fourier Series? 58 314.2 The Fourier Series of a Function 58 6
14 2 1 Even and Odd Functions 58914.3 Convergence of Fourier Series 59 3
14 3 1 Convergence at the End Points 59 914.3 2 A Second Convergence Theorem 60 114.3 3 Partial Sums of Fourier Series 60414.3 4 The Gibbs Phenomenon 606
14 4 Fourier Cosine and Sine Series 609
14 4 1 The Fourier Cosine Series of a Function 61 0
14 4 2 The Fourier Sine Series of a Function 61 2
14 5 Integration and Differentiation of Fourier Series 61 4
14 6 The Phase Angle Form of a Fourier Series 62 3
14 7 Complex Fourier Series and the Frequency Spectrum 63 0
14 7 1 Review of Complex Numbers 63 0
14 7 2 Complex Fourier Series 631
Trang 9Chapter 15 The Fourier Integral and Fourier Transforms 63 7
15 1 The Fourier Integral 637
15 2 Fourier Cosine and Sine Integrals 64 0
15 3 The Complex Fourier Integral and the Fourier Transform 64 2
15 4 Additional Properties and Applications of the Fourier Transform 65 2
15 4.1 The Fourier Transform of a Derivative 65 2
15 4.2 Frequency Differentiation 65 5
15 4 3 The Fourier Transform of an Integral 65 6
15 4 4 Convolution 65 7
15 4 5 Filtering and the Dirac Delta Function 66 0
15 4 6 The Windowed Fourier Transform 66 1
15 4 7 The Shannon Sampling Theorem 66 5
15 4 8 Lowpass and Bandpass Filters 66 7
15 5 The Fourier Cosine and Sine Transforms 67 0
15 6 The Finite Fourier Cosine and Sine Transforms 67 3
15 7 The Discrete Fourier Transform 67 5
15 7 1 Linearity and Periodicity 678
15 7 2 The Inverse N-Point DFT 67 8
15 7 3 DFT Approximation of Fourier Coefficients 67 9
15 8 Sampled Fourier Series 68 1
15 8 1 Approximation of a Fourier Transform by an N-Point DFT 685
15 8 2 Filtering 68 9
15 9 The Fast Fourier Transform 69 4
15 9 1 Use of the FFT in Analyzing Power Spectral Densities of Signals 69 5
15 9 2 Filtering Noise From a Signal 69 6
15 9 3 Analysis of the Tides in Morro Bay 69 7
Chapter 16 Special Functions, Orthogonal Expansions, and Wavelets 70 1
16 1 Legendre Polynomials 70 1
16 1 1 A Generating Function for the Legendre Polynomials 704
16 1 2 A Recurrence Relation for the Legendre Polynomials 70 6
16 1 3 Orthogonality of the Legendre Polynomials 70 8
16 1 4 Fourier-Legendre Series 70 9
16 1 5 Computation of Fourier-Legendre Coefficients 71 1
16 1 6 Zeros of the Legendre Polynomials 71 3
16 1 7 Derivative and Integral Formulas for Pn (x) 71 5
16 2 Bessel Functions 71 9
16 2.1 The Gamma Function 71 9
16 2.2 Bessel Functions of the First Kind and Solutions of Bessel 's Equation 72 1
16 2.3 Bessel Functions of the Second Kind 72 2
16 2.4 Modified Bessel Functions 72 5
16 2.5 Some Applications of Bessel Functions 72 7
16 2.6 A Generating Function for L(x) 73 2
16 2.7 An Integral Formula forL(x) 73 3
16 2.8 A Recurrence Relation for Jv (x) 73 5
16 2.9 Zeros of Jv (x) 737
Trang 1016 2 10 Fourier-Bessel Expansions 73 9
16 2 11 Fourier-Bessel Coefficients 74 1
16 3 Sturm-Liouville Theory and Eigenfunction Expansions 74 5
16 3 1 The Sturm-Liouville Problem 74 516.3 2 The Sturm-Liouville Theorem 75 2
16 3 3 Eigenfunction Expansions 75 5
16 3 4 Approximation in the Mean and Bessel ' s Inequality 75 9
16 3 5 Convergence in the Mean and Parseval 's Theorem 76 2
16 3 6 Completeness of the Eigenfunctions 76 3
16 4 Wavelets 765
16 4 1 The Idea Behind Wavelets 765
16 4 2 The Haar Wavelets 76 7
16 4 3 A Wavelet Expansion 77 4
16 4 4 Multiresolution Analysis with Haar Wavelets 77 4
16 4 5 General Construction of Wavelets and Multiresolution Analysis 77 5
16 4.6 Shannon Wavelets 77 6
PART 6 Partial Differential Equations 779
Chapter 17 The Wave Equation 78 1
17 1 The Wave Equation and Initial and Boundary Conditions 78 1
17 2 Fourier Series Solutions of the Wave Equation 78 6
17 2 1 Vibrating String with Zero Initial Velocity 78 6
17 2 2 Vibrating String with Given Initial Velocity and Zero Initial Displacement 79 1 17 2 3 Vibrating String with Initial Displacement and Velocity 79 3
17 2 4 Verification of Solutions 79 4
17 2 5 Transformation of Boundary Value Problems Involving the Wave Equation 79 6
17 2 6 Effects of Initial Conditions and Constants on the Motion 79 8
17 2 7 Numerical Solution of the Wave Equation 80 1
17 3 Wave Motion Along Infinite and Semi-Infinite Strings 80 8
17 3 1 Wave Motion Along an Infinite String 80 8
17 3 2 Wave Motion Along a Semi-Infinite String 81 3
17 3 3 Fourier Transform Solution of Problems on Unbounded Domains 81 5
17 4 Characteristics and d'Alembert' s Solution 822
17 4 1 A Nonhomogeneous Wave Equation 82 5
17 4 2 Forward and Backward Waves 82 8
17 5 Normal Modes of Vibration of a Circular Elastic Membrane 83 1
17 6 Vibrations of a Circular Elastic Membrane, Revisited 83 4
17 7 Vibrations of a Rectangular Membrane 83 7
Chapter 18 The Heat Equation 84 1
18 1 The Heat Equation and Initial and Boundary Conditions 84 1
18 2 Fourier Series Solutions of the Heat Equation 844
Trang 1118 2 1 Ends of the Bar Kept at Temperature Zero 844
18 2 2 Temperature in a Bar with Insulated Ends 84 7
18 2 3 Temperature Distribution in a Bar with Radiating End 84 8
18 2 4 Transformations of Boundary Value Problems Involving the Heat Equation 85 1
18 2 5 A Nonhomogeneous Heat Equation 85 418.2 6 Effects of Boundary Conditions and Constants on Heat Conduction 85 7
18 2 7 Numerical Approximation of Solutions 85 918.3 Heat Conduction in Infinite Media 86 5
18 3 1 Heat Conduction in an Infinite Bar 865
18 3 2 Heat Conduction in a Semi-Infinite Bar 86 8
18 3 3 Integral Transform Methods for the Heat Equation in an Infinite Medium 869
18 4 Heat Conduction in an Infinite Cylinder 87 3
18 5 Heat Conduction in a Rectangular Plate 87 7
Chapter 19 The Potential Equation 879
19 1 Harmonic Functions and the Dirichlet Problem 87 9
19 2 Dirichlet Problem for a Rectangle 88 1
19 3 Dirichlet Problem for a Disk 88 3
19 4 Poisson' s Integral Formula for the Disk 886
19 5 Dirichlet Problems in Unbounded Regions 88 8
19 5 1 Dirichlet Problem for the Upper Half Plane 88 9
19 5 2 Dirichlet Problem for the Right Quarter Plane 89 1
19 5 3 An Electrostatic Potential Problem 89 3
19 6 A Dirichlet Problem for a Cube 89 6
19 7 The Steady-State Heat Equation for a Solid Sphere 89 8
19 8 The Neumann Problem 902
19 8 1 A Neumann Problem for a Rectangle 90 4
19 8 2 A Neumann Problem for a Disk 90 6
19 8 3 A Neumann Problem for the Upper Half Plane 90 8
Chapter 20 Geometry and Arithmetic of Complex Numbers 91 3
20 1 Complex Numbers 91 3
20 1 1 The Complex Plane 91 4
20 1 2 Magnitude and Conjugate 91 5
20 2.2 Circles and Disks 92 2
20 2.3 The Equation lz -al = Iz - bI 923
20 2.4 Other Loci 925
20 2.5 Interior Points, Boundary Points, and Open and Closed Sets 925
Trang 12Contents xiii
20 2 6 Limit Points 929
20 2 7 Complex Sequences 93 1
20 2 8 Subsequences 93 4
20 2 9 Compactness and the Bolzano-Weierstrass Theorem 93 5
Chapter 21 Complex Functions 93 9
21 1 Limits, Continuity, and Derivatives 93 9
21 1 1 Limits 93 9
21 1 2 Continuity 94 1
21 1 3 The Derivative of a Complex Function 94 3
21 1 4 The Cauchy-Riemann Equations 94 5
21 2 Power Series 95 0
21 2 1 Series of Complex Numbers 95 1
21 2.2 Power Series 95 2
21 3 The Exponential and Trigonometric Functions 95 7
21 4 The Complex Logarithm 966
Chapter 22 Complex Integration 97 5
22 1 Curves in the Plane 97 5
22 2 The Integral of a Complex Function 98 0
22 2 1 The Complex Integral in Terms of Real Integrals 98 3
22 2 2 Properties of Complex Integrals 985
22 2 3 Integrals of Series of Functions 98 822.3 Cauchy' s Theorem 99 0
22 3 1 Proof of Cauchy' s Theorem for a Special Case 99 3
22 4 Consequences of Cauchy's Theorem 99 4
22 4 1 Independence of Path 994
22 4 2 The Deformation Theorem 99 5
22 4 3 Cauchy's Integral Formula 99 7
22 4 4 Cauchy's Integral Formula for Higher Derivatives 100 0
22 4 5 Bounds on Derivatives and Liouville ' s Theorem 100 1
22 4 6 An Extended Deformation Theorem 100 2
Chapter 23 Series Representations of Functions 100 7
23 1 Power Series Representations 100 7
23 1 1 Isolated Zeros and the Identity Theorem 101 2
23 1 2 The Maximum Modulus Theorem 101 6
23 2 The Laurent Expansion 101 9
Chapter 24 Singularities and the Residue Theorem 102 3
24 1 Singularities 1023
24 2 The Residue Theorem 103 0
24 3 Some Applications of the Residue Theorem 1037
Trang 1324 3 1 The Argument Principle 103 7
24 3 2 An Inversion for the Laplace Transform 103 9
24 3 3 Evaluation of Real Integrals 1040
Chapter 25 Conformal Mappings 105 5
25 1 Functions as Mappings 105 5
25 2 Conformal Mappings 106 2
25 2 1 Linear Fractional Transformations 106 4
25 3 Construction of Conformal Mappings Between Domains 107 2
25 3 1 Schwarz-Christoffel Transformation 107 7
25 4 Harmonic Functions and the Dirichlet Problem 108 0
25 4 1 Solution of Dirichlet Problems by Conformal Mapping 108 3
25 5 Complex Function Models of Plane Fluid Flow 108 7
Chapter 26 Counting and Probability 109 9
26 1 The Multiplication Principle 109 9
26 2 Permutations 1102
26 3 Choosing r Objects from n Objects 1104
26 3 1 r Objects from n Objects, with Order 110 4
26 3 2 r Objects from n Objects, without Order 110 6
26 3 3 Tree Diagrams 1107
26 4 Events and Sample Spaces 111 2
26 5 The Probability of an Event 111 6
26 6 Complementary Events 112 1
26 7 Conditional Probability 112 2
26 8 Independent Events 112 6
26 8 1 The Product Rule 112 8
26 9 Tree Diagrams in Computing Probabilities 113 0
27 2 Random Variables and Probability Distributions 1150
27 3 The Binomial and Poisson Distributions 1154
27 3 1 The Binomial Distribution 1154
27 3 2 The Poisson Distribution 115 7
27 4 A Coin Tossing Experiment, Normally Distributed Data, and the Bell Curve 115 9
27 4 1 The Standard Bell Curve 117 4
27 4 2 The 68, 95, 99 7 Rule 1176
Trang 1427 5 Sampling Distributions and the Central Limit Theorem 117 827.6 Confidence Intervals and Estimating Population Proportion 118 5
27 7 Estimating Population Mean and the Student t Distribution 1190
27 8 Correlation and Regression 1194
Answers and Solutions to Selected Problems Al
Index 1 1
Trang 16This Sixth Edition ofAdvanced Engineering Mathematicsmaintains the primary goal of ous editions-to engage much of the post-calculus mathematics needed and used by scientists ,engineers, and applied mathematicians, in a setting that is helpful to both students and faculty The format used throughout begins with the correct developments of concepts such as Fourie rseries and integrals, conformal mappings, and special functions These ideas are 'then brought tobear on applications and models of important phenomena, such as wave and heat propagatio nand filtering of signals
previ-This edition differs from the previous one primarily in the inclusion of statistics an dnumerical methods The statistics part treats random variables, normally distributed data, bel lcurves, the binomial, Poisson, and student t-distributions, the central limit theorem, confidenc eintervals, correlation, and regression This is preceded by prerequisite topics from probabilityand techniques of enumeration
The numerical methods are applied to initial value problems in ordinary differential tions, including a proposal for radioactive waste disposal, and to boundary value problem sinvolving the heat and wave equations
equa-Finally, in order to include these topics without lengthening the book, some items from th efifth edition have been moved to a website, located at http ://engineering.thornsonlearning com
I hope that this provides convenient accessibility Material selected for this move includessome biographies and historical notes, predator/prey and competing species models, the theor yunderlying the efficiency of the FFT, and some selected examples and problems
The chart on the following page offers a complete organizational overview
Acknowledgments
This book is the result of a team effort involving much more than an author Among those
to whom I owe a debt of appreciation are Chris Carson, Joanne Woods, Hilda Gowans an dKamilah Reid-Burrell of Thomson Engineering, and Rose Kernan and the professionals at RP KEditorial Services, Inc I also want to thank Dr Thomas O'Neil of the California Polytechni cState University for material he contributed, and Rich Jones, who had the vision for the firs tedition of this book many years ago
Finally, I want to acknowledge the reviewers, whose suggestions for improvements an dclarifications are much appreciated :
Preliminary Review
Panagiotis Dimitrakopoulos, University of Marylan d
Mohamed M Hafez, University of California, Davi s
Jennifer Hopwood, University of Western Australia
Nun Kwan Yip, Purdue University
Trang 17Ordinary Differential Vectors, Matrices,
Systems of
Transforms Solutions Functions Expansions ,
Completeness
Qualitative Methods, / Stability, Analysis of
Critical Points
Probability
IStatistics
Fourier Analysis
e r Fourier Series, Fourier Discrete Fouri
Integrals
Statistical Analysis
Trang 18Preface xix Draft Revie w
Sabri Abou-Ward, University of Toront o
Craig Hildebrand, California State University - Fresn o
Seiichi Nomura, University of Texas, Arlingto n
David L Russell, Virginia Polytechnic Institute and State University
Y Q Sheng, McMaster University
PETER V O ' NEI L
University of Alabama at Birmingham
Trang 20A differential equation is an equation that contains one or more derivatives For example,
y"(x) + y(x) = 4 sin(3x )and
4
d w _ (w ( t)) 2=e - `
are differential equations These are ordinary differential equations because they involve onl ytotal derivatives, rather than partial derivatives
Differential equations are interesting and important because they express relationship sinvolving rates of change Such relationships form the basis for developing ideas and studyin gphenomena in the sciences, engineering, economics, and increasingly in other areas, such as th ebusiness world and the stock market We will see examples of applications as we learn moreabout differential equations
Ordinary Differential Equations
dt 4
Trang 21y " +4y = -4 sin(2x) +4 sin(2x) = 0 This solution is defined for all x (that is, on the whole real line)
By contrast,
y=xln(x)- x
is a solutionof
Y =y + 1 ,xbut this solution is defined only for x > 0 Indeed, the coefficient 1/x of y in this equatio nmeans that x 0 is disallowed from the start
We now begin a systematic development of ordinary differential equations, starting wit hthe first order case
2
Trang 22PRELIMINARY ONCEPTS SEPARABLE EQUATION S
i i O,k '10( t ,NJ U7, BERNOULLI, AND l 1ICC ; tr41 El9LJl n
T ONS APPLICATIONS TO MECHANICS, ELECTRICA LCIRCUITS , AND ORTHOGONAL TRAJECTORIES E N
Before developing techniques for solving various kinds of differential equations, we will develo p
some terminology and geometric insight
1 1 1 General and Particular Solutions
A first-order differential equation is any equation involving a first derivative, but no highe r
derivative In its most general form, it has the appearanc e
Note that y'must be present for an equation to qualify as a first-order differential equation, bu t
x and/or y need not occur explicitly
A solutionof equation (1 1) on an interval Iis a function cp that satisfies the equation forallx inI That is,
F(x, cp (x) , co' (x)) =0 for allxin I For example,
cp (x) = 2 + ke X
3
Trang 23for all x > 0, and for any number c
In both of these examples, the solution contained an arbitrary constant This is a symbolindependent of x and y that can be assigned any numerical value Such a solution is called th e
general solution of the differential equation Thu s
cp(x) = 2 + ke -X
is the general solution of y' + y = 2
Each choice of the constant in the general solution yields a particular solution For example,
f(x) = 2 + g (x) = 2 - e - X
and
h(x) = 2- 53e-x
are all particular solutions of y' + y = 2, obtained by choosing, respectively, k = 1, -1 and
3 in the general solution
- N/ j.
1 1 2 Implicitly Defined Solution s
Sometimes we can write a solution explicitly giving y as a function of x For example ,
y = ke -x
is the general solution of
y = -Y,
as can be verified by substitution This general solution is explicit, with y isolated on one side
of an equation, and a function of x on the other
By contrast, consider
2xy3+ 2Y
3x2 y2+ 8e4Y
We claim that the general solution is the function y(x) implicitly defined by the equatio n
in which k can be any number To verify this, implicitly differentiate equation (1 2) with respec t
to x, remembering that y is a function of x We obtain
2xy3+ 3x2 y2y' + 2 + 8 e4Y y ' = 0 ,and solving for y' yields the differential equation
In this example we are unable to solve equation (1 2) explicitly for y as a function of x,isolating y on one side Equation (1 2), implicitly defining the general solution, was obtaine d
by a technique we will develop shortly, but this technique cannot guarantee an explicit solution
Trang 241 1 Preliminary Concept s
1 1 3 Integral Curves
A graph of a solution of a first-order differential equation is called an integral curve of theequation If we know the general solution, we obtain an infinite family of integral curves, on efor each choice of the arbitrary constant
for all x The integral curves of y' + y = 2 are graphs of y =2 + ke-x for different choices of
k Some of these are shown in Figure 1 1
Y
3 0 k=
2 0 k=
is
Trang 25for x 0 Graphs of some of these integral curves, obtained by making choices for c, are show n
EXAMPLE 1 3
The differential equation
y ' +xy= 2has general solution
Y( x ) = e -x212fx 2,12 dk -F ke _x2/2
0
Figure 1 3 shows computer-generated integral curves corresponding to k = 0, 4, 13, -7, -1 5and -11
1 1 4 The Initial Value Proble m
The general solution of a first-order differential equation F(x, y, y') = 0 contains an arbitrary
constant, hence there is an infinite family of integral curves, one for each choice of the constant
If we specify that a solution is to pass through a particular point (xo, yo), then we must find that
particular integral curve (or curves) passing through this point This is called an initial valu e
problem Thus, a first order initial value problem has the form
F(x, Y, Y) =0 ; Y(xo) = Yo ,
in which xo and yo are given numbers The condition y(xo) = yo is called an initial condition
Trang 26Asa check, y(1) =2-7 = -5
The effect of the initial condition in this example was to pick out one special integral curv e
as the solution sought This suggests that an initial value problem may be expected to have aunique solution We will see later that this is the case, under mild conditions on the coefficient s
in the differential equation
1 1.5 Direction Fields
Imagine a curve, as in Figure 1 4 If we choose some points on the curve and, at each point ,draw a segment of the tangent to the curve there, then these segments give a rough outline o fthe shape of the curve This simple observation is the key to a powerful device for envisionin gintegral curves of a differential equation
Trang 27FIGURE 1 4 Short tangen t segments suggest the shap e
of the curve.
The general first-order differential equation has the for m
F(x,y,y')=0 Suppose we can solve for y' and write the differential equation a s
y' = f( x , Y) •
Here f is a known function Suppose f(x, y) is defined for all points (x, y) in some region
R of the plane The slope of the integral curve through a given point (xo, yo) of R is y'(xo) ,
which equals f(xo,yo) If we compute f(x, y) at selected points in R, and draw a small line segment having slope f(x, y) at each (x, y), we obtain a collection of segments which trace out
the shapes of the integral curves This enables us to obtain important insight into the behavio r
of the solutions (such as where solutions are increasing or decreasing, limits they might hav e
at various points, or behavior as x increases)
A drawing of the plane, with short line segments of slope f(x,y) drawn at selected point s
(x,y),iscalleda direction field of the differential equation y' = f(x,A The name derives fro mthe fact that at each point the line segment gives the direction of the integral curve through tha tpoint The line segments are called lineal elements
is shown in Figure 1 5(a) The lineal elements form a profile of some integral curves and giv e
us some insight into the behavior of solutions, at least in this part of the plane Figure 1 5(b )reproduces this direction field, with graphs of the integral curves through (0, 1), (0, 2), (0, 3) ,(0, -1), (0, -2) and (0, -3)
By a method we will develop, the general solution of y' = y 2i s
1
so the integral curves form a family of hyperbolas, as suggested by the curves sketched i nFigure 1 5(b)
Trang 29EXAMPLE 1 6
Figure 1 6 shows a direction field for
y' = sin(xy) ,together with the integral curves through (0, 1), (0, 2), (0, 3), (0, -1), (0, -2) and (0, -3) Inthis case, we cannot write a simple expression for the general solution, and the direction fiel dprovides information about the behavior of solutions that is not otherwise readily apparent
first-In each of Problems 1 through 6, determine whether th e
given function is a solution of the differential equation
7 y2 + xy - 2x2 - 3x - 2y = C;
y-4x-2+(x+2y-2)y'=0
Trang 301 2 Separable Equations 1 1
8 xy3-y= C;y3 +(3xy2 -1)y'= 0
9 y 2 - 4x 2 + ex➢ = C; 8x - yexY - (2y + xex}' )Y ' = 0
0
In each of Problems 12 through 16, solve the initial valu e
problem and graph the solution Hint: Each of these
dif-ferential equations can be solved by direct integration Us e
the initial condition to solve for the constant of integration
In each of Problems 17 through 20 draw some linea l
elements of the differential equation for -4 < x < 4,
-4 < y < 4 Use the resulting direction field to sketch a
graph of the solution of the initial value problem (Thes e
problems can be done by hand )
DEFINITION 1.1 Separable Differential Equation
A differential equationis called separable if it can be writte n
Trang 311 =-e -x +k , Y
an equation that implicitly defines the general solution In this example we can explicitly solv efor y, obtaining the general solution
1
Y _
e -x - k
Now recall that we required that y 0 in order to separate the variables by dividing byy2 In fact, the zero function y(x) = 0 is a solution of y'= y2ex , although it cannot be obtaine dfrom the general solution by any choice ofk : For this reason, y(x) = 0 is called a singularsolution of this equation
Figure 1 7 shows graphs of particular solutions obtained by choosing k as 0, 3, -3, 6 an d-6
FIGURE 1 7 Integral curves of y ' = y2 e -x fo r k=0,3,-3,6, and -6.
Whenever we use separation of variables, we must be alert to solutions potentially los tthrough conditions imposed by the algebra used to make the separation
as
Trang 32and y = -1 into the differential equation to obtain the correct equation 0 = O
Now integrate the separated equation to obtai n
1n11+YJ= 1
1
+k.
This implicitly defines the general solution In this case, we can solve for y(x) explicitly Begi n
by taking the exponential of both sides to obtain
in which B is any nonzero number
Now revisit the assumption that,x 0 and y -1 In the general solution, we actuallyobtain y = -1 if we allow B = O Further, the constant function y(x) = -1 does satisfy
x2 y'= 1 +y Thus, by allowingB to be any number, including 0, the general solution
Trang 33k=e-u- 41
.The solution of the initial value problem i s
1 1y+31nIYI = 3(x - 1 ) 3 -
1 2 1 Some Applications of Separable Differential Equation s
Separable equations arise in many contexts, of which we will discuss three
e-x+4 -e-I
Trang 34to decrease Assuming (for want of better information) that the victim's temperature was a
"normal" 98 6 at the time of death, the lieutenant will try to estimate this time by observing th ebody's current temperature and calculating how long it would have had to lose heat to reachthis point
According to Newton's law of cooling, the body will radiate heat energy into the room a t
a rate proportional to the difference in temperature between the body and the room If T(t) i s the body temperature at time t, then for some constant of proportionality k,
T' (t) = k [T(t) - 68]
The lieutenant recognizes this as a separable differential equation and write s
1dT=kdt
Now the constants k and B must be determined, and this requires information The lieutenant
arrived at 9 :40 p m and immediately measured the body temperature, obtaining 94 4 degrees Letting 9 :40 be time zero for convenience, this means tha t
T(0) = 94 4 = 68 + B , and so B = 26 4 Thus far,
T(t) = 68+26 4e k'
To determine k, the lieutenant makes another measurement At 11 :00 she finds that thebody temperature is 89 2 degrees Since 11 :00 is 80 minutes past 9 :40, this means that
T(80) = 89 2 = 68 + 26 4e s°k.Then
e sok= 21 2
(21 280k =1n
26 4
T-6 8Upon integrating, she gets
Taking exponentials, she gets
in which A = e c Then
Then
26 4 'so
Trang 35k= 1 In 21 2
80 26 4The lieutenant now has the temperature function :
30 6
t ln(21 2 1
26 4 80 1\\26 4))
tln(21 2/26 4) 'which is approximately -53 8 minutes Death occurred approximately 53 8 minutes before(because of the negative sign) the first measurement at 9 :40, which was chosen as time zero This puts the murder at about 8 :46 p m
This is a separable differential equation Write it a s
and integrate to obtain
Since mass is positive, Iml =m and
801n(30 6/26 4)
Trang 361 2 Separable Equations 17
Determination of A and k for a given element requires two measurements Suppose at
some time, designated as time zero, there are M grams present This is called the initial mass Then
We obtain a more convenient formula for the mass if we choose the time of the secon d
measurement more carefully Suppose we make the second measurement at that time T = H
at which exactly half of the mass has radiated away At this time, half of the mass remains, s o
M T = M/2 and M T /M = 1/2 Now the expression for the mass become s
m(t) = Meln(1/2)t/ x
or
7 2 (t) = Me- in (2)t/H
This number H is called the half-life of the element Although we took it to be the tim e
needed for half of the original amount M to decay, in fact, between any times t1 and t1+H ,
exactly half of the mass of the element present at tl will radiate away To see this, writ e
some of it into radioactive carbon-14, or 14 C This element has a half-life of about 5,730 years
Over the relatively recent period of the history of this planet in which life has evolved, th efraction of 14 C in the atmosphere, compared to regular carbon, has been essentially constant This means that living matter (plant or animal) has injested 14 C at about the same rate over along historical period, and objects living, say, two million years ago would have had the sam eratio of carbon-14 to carbon in their bodies as objects alive today When an organism dies, itceases its intake of 14 C, which then begins to decay By measuring the ratio of 14 C to carbon
in an artifact, we can estimate the amount of the decay, and hence the time it took, giving a nhence
Trang 37estimate of the time the organism was alive This process of estimating the age of an artifac t
is called carbon dating Of course, in reality the ratio of 14 C in the atmosphere has only been
approximately constant, and in addition a sample may have been contaminated by exposure t oother living organisms, or even to the air, so carbon dating is a sensitive process that can lea d
to controversial results Nevertheless, when applied rigorously and combined with other test sand information, it has proved a valuable tool in historical and archeological studies
To apply equation (1 4) to carbon dating, use H = 5730 and comput e
Now suppose we have an artifact, say a piece of fossilized wood, and measurements show tha t
the ratio of 14 C to carbon in the sample is 37 percent of the current ratio If we say that th e
wood died at time 0, then we want to compute the time T it would take for one gram of th eradioactive carbon to decay this amount Thus, solve for T in
0 37 = e-0.000120968r
We find that
ln(0 37 )
T _ 0.000120968 8 '21 9years This is a little less than one and one-half half-lives, a reasonable estimate if nearly s ofthe 14C has decayed
EXAMPLE 1 1 3
(Torricelli's Law) Suppose we want to estimate how long it will take for a container to empt y
by discharging fluid through a drain hole This is a simple enough problem for, say, a sod acan, but not quite so easy for a large oil storage tank or chemical facility
We need two principles from physics The first is that the rate of discharge of a fluidflowing through an opening at the bottom of a container is given b y
dV
= -kAv ,
d t
in which V(t) is the volume of fluid in the container at time t, v(t) is the discharge velocity
of fluid through the opening, A is the cross sectional area of the opening (assumed constant) , and k is a constant determined by the viscosity of the fluid, the shape of the opening, and th e
fact that the cross-sectional area of fluid pouring out of the opening is slightly less than tha t
of the opening itself In practice, k must be determined for the particular fluid, container, an d
opening, and is a number between 0 and 1
We also need Torricelli's law, which states that v(t) is equal to the velocity of a free-fallin g
particle releasedfroma height equal to the depth of the fluid at time t (Free-falling mean s
that the particle is influenced by gravity only) Now the work done by gravity in moving the
particle from its initial point by a distance h(t) is nigh(t), and this must equal the change in
the kinetic energy, (2)mv 2 Therefore,
v(t) = A /2gh(t)
Trang 38We will apply equation (1 5) to a specific case to illustrate its use Suppose we have a
hemispherical tank of water, as in Figure 1 9 The tank has radius 18 feet, and water drains through a circular hole of radius 3 inches at the bottom How long will it take the tank to
empty ?
Equation (1 5) contains two unknown functions, V(t) and h(t), so one must be eliminated Let r(t) be the radius of the surface of the fluid at time t and consider an interval of time from to to t, = to + At The volume AV of water draining from the tank in this time equals th e volume of a disk of thickness Ah (the change in depth) and radius r(t*), for some t* betwee n
Take g to be 32 feet per second per second The radius of the circular opening is 3 inches, or
a feet, so its area is A = ar/16 square feet For water, and an opening of this shape and size ,the experiment gives k = 0 8 The last equation becomes
Trang 39The last three examples contain an important message Differential equations can be used
to solve a variety of problems, but a problem usually does not present itself as a differentia lequation Normally we have some event or process, and we must use whatever informatio n
we have about it to derive a differential equation and initial conditions This process is called
mathematical modeling The model consists of the differential equation and other relevan t
information, such as initial conditions We look for a function satisfying the differential equationand the other information, in the hope of being able to predict future behavior, or perhaps betterunderstand the process being considered
In each of Problems 1 through 10, determine if the
dif-ferential equation is separable If it is, find the general
solution (perhaps implicitly defined) If it is not
separa-ble, do not attempt a solution at this time
15 yy' = 2x sec(3y) ; y(2/3) = 7r/3
16 An object having a temperature of 90 degrees heit is placed into an environment kept at 60 degrees Ten minutes later the object has cooled to 88 degrees What will be the temperature of the object after it ha sbeen in this environment for 20 minutes? How lon gwill it take for the object to cool to 65 degrees ?
Fahren-17 A thermometer is carried outside a house whose bient temperature is 70 degrees Fahrenheit After fiveminutes the thermometer reads 60 degrees, and fiftee nminutes after this, 50 4 degrees What is the outsidetemperature (which is assumed to be constant) ?
am-8
Trang 401 2 Separable Equations 21
18 Assume that the population of bacteria in a petri dis h
changes at a rate proportional to the population a t
that time This means that, ifP(t) isthe population
at timet, then
dP
= kP dt
for some constantk Aparticular culture has a
popu-lation density of 100,000 bacteria per square inch A
culture that covered an area of 1 square inch at 10 :0 0
a m on Tuesday was found to have grown to cover 3
square inches by noon the following Thursday Ho w
many bacteria will be present at 3 :00 p.m the
follow-ing Sunday? How many will be present on Monday a t
4 :00 p.m ? When will the world be overrun by thes e
bacteria, assuming that they can live anywhere on th e
earth's surface? (Here you need to look up the lan d
area of the earth )
19 Assume that a sphere of ice melts at a rate
pro-portional to its surface area, retaining a spherical
shape Interpret melting as a reduction of volume wit h
respect to time Determine an expression for the
vol-ume of the ice at any time t
20 A radioactive element has a half-life of ln(2) weeks
Ife 3 tons are present at a given time, how much wil l
be left 3 weeks later ?
21 The half-life of uranium-238 is approximately
4.5 years How much of a 10-kilogram bloc k
of U-238 will be present 1 billion years from now ?
22 Given that 12 grams of a radioactive element decays
to 9 1 grams in 4 minutes, what is the half-life of thi s
Calculate I'(x) by differentiating under the integral
sign, then letu=x/t Show that I'(x) =-2I(x) and
solve for I(x) Evaluate the constant by using the
stan-dard result that foe-` 2 dt = J/2 Finally, evaluate
1(3)
24 Derive the fact used in Example 1 13 that v(t) =
,/2gh(t) Hint : Consider a free-falling particle
hav-ing heighth(t)at timet The work done by gravity in
moving the particle from its starting point to a given
point isnigh(t),and this must equal the change in the
kinetic energy, which is (1/2)nzv2
25 Calculate the time required to empty the
hemispher-ical tank of Example 1 13 if the tank is positione dwith its flat side down
26 (Draining a Hot Tub) Consider a cylindrical hot tu b
with a 5-foot radius and height of 4 feet, placed o none of its circular ends Water is draining from the tubthrough a circular holes inches in diameter locate d
in the base of the tub (a) Assume a value k = 0 6 to determine the rate atwhich the depth of the water is changing Here it isuseful to write
dh dh dV _ dV/dt _= =
.dt dV dt dV/d h
(b) Calculate the time T required to drain the hot tub i f
it is initially full Hint:One way to do this is to write
T= f dt dh
H dh
(c) Determine how much longer it takes to drain th elower half than the upper half of the tub Hint : Us ethe integral suggested in (b), with different limits fo rthe two halves
27 (Draining a Cone) A tank shaped like a right circula rcone, with its vertex down, is 9 feet high and has adiameter of 8 feet It is initially full of water (a) Determine the time required to drain the tankthrough a circular hole of diameter 2 inches at th evertex Takek =0 6
(b) Determine the time it takes to drain the tank if i t
is inverted and the drain hole is of the same size an dshape as in (a), but now located in the new base
28 (Drain Hole at Unknown Depth) Determine the rate ofchange of the depth of water in the tank of Proble m
27 (vertex at the bottom) if the drain hole is located i nthe side of the cone 2 feet above the bottom of the tank What is the rate of change in the depth of the water whe nthe drain hole is located in the bottom of the tank? Is i tpossible to determine the location of the drain hole if w eare told the rate of change of the depth and the depth ofthe water in the tank? Can this be done without knowin gthe size of the drain opening?
29 Suppose the conical tank of Problem 27, vertex at th ebottom, is initially empty and water is added at th econstant rate ofar/10cubic feet per second Does th etank ever overflow?
30 (Draining a Sphere) Determine the time it takes t ocompletely drain a spherical tank of radius 18 feet if
it is initially full of water and the water drains through
a circular hole of radius 3 inches located in the botto m
of the tank Usek =0 8