10th Edition | ISBN: 9780470458365 / 0470458364

3,663

expert-verified solutions in this book

10th Edition | ISBN: 9780470458365 / 0470458364

3,663

expert-verified solutions in this book

### Chapter 1

First-Order Odes

 1.1 Basic Concepts. Modeling Problem Set p.8 1.2 Geometric Meaning of y'=f(x,y). Direction Fields, Euler's Method Problem Set p.11 1.3 Separable ODEs. Modeling Problem Set p.18 1.4 Exact ODEs. Integrating Factors Problem Set p.26 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics Problem Set p.34 1.6 Orthogonal Trajectories Problem Set p.38 1.7 Existence and Uniqueness of Solutions for Initial Value Problems Problem Set p.42 Review Questions and Problems p.43

### Chapter 2

Second-Order Linear Odes

 2.1 Homogeneous Linear ODEs of Second Order Problem Set p.53 2.2 Homogeneous Linear ODEs with Constant Coefficients Problem Set p.59 2.3 Differential Operators Problem Set p.61 2.4 Modeling of Free Oscillations of a Mass-Spring System Problem Set p.69 2.5 Euler-Cauchy Equations Problem Set p.73 2.6 Existence and Uniqueness of Solutions. Wronskian Problem Set p.79 2.7 Nonhomogeneous ODEs Problem Set p.84 2.8 Modeling: Forced Oscillations. Resonance Problem Set p.91 2.9 Modeling: Electric Circuits Problem Set p.98 2.10 Solution by Variation of Parameters Problem Set p.102 Review Questions and Problems p.102

### Chapter 3

Higher Order Linear Odes

 3.1 Homogeneous Linear ODEs Problem Set p.111 3.2 Homogeneous Linear ODEs with Constant Coefficients Problem Set p.116 Review Questions and Problems p.122 3.3 Nonhomogeneous Linear ODEs Problem Set p.122

### Chapter 4

Systems Of Odes. Phase Plane. Qualitative Methods

 4.1 Systems of ODEs as Models in Engineering Applications Problem Set p.136 4.3 Constant-Coefficient Systems. Phase Plane Method Problem Set p.147 4.4 Criteria for Critical Points. Stability Problem Set p.151 4.5 Qualitative Methods for Nonlinear Systems Problem Set p.159 4.6 Nonhomogeneous Linear Systems of ODEs Problem Set p.163 Review Questions and Problems p.164

### Chapter 5

Series Solutions Of Odes. Special Functions

 5.1 Power Series Method Problem Set p.174 5.2 Legendre's Equation. Legendre Polynomials Pn(x) Problem Set p.179 5.3 Extended Power Series Method: Frobenius Method Problem Set p.186 5.4 Bessel's Equation. Bessel Functions Jv(x) Problem Set p.195 5.5 Bessel Functions Yv(x). General Solution Problem Set p.200 Review Questions and Problems p.200

### Chapter 6

Laplace Transforms

 6.1 Laplace Transform. Linearity. First Shifting Theorem (s-Shifting) Problem Set p.210 6.2 Transforms of Derivatives and Integrals. ODEs Problem Set p.216 6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-Shifting) Problem Set p.223 6.4 Short Impulses. Dirac's Delta Function. Partial Fractions Problem Set p.230 6.5 Convolution. Integral Equations Problem Set p.237 6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients Problem Set p.241 6.7 Systems of ODEs Problem Set p.246 Review Questions and Problems p.251

### Chapter 7

Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

 7.1 Matrices, Vectors: Addition and Scalar Multiplication Problem Set p.261 7.2 Matrix Multiplication Problem Set p.270 7.3 Linear Systems of Equations. Gauss Elimination Problem Set p.280 7.4 Linear Independence. Rank of a Matrix. Vector Space Problem Set p.287 7.7 Determinants. Cramer's Rule Problem Set p.300 7.8 Inverse of a Matrix. Gauss-Jordan Elimination Problem Set p.308 7.9 Vector Spaces, Inner Product Spaces, Linear Transformations Problem Set p.318 Review Questions and Problems p.318

### Chapter 8

Linear Algebra: Matrix Eigenvvalue Problems

 8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors Problem Set p.329 8.2 Some Applications of Eigenvalue Problems Problem Set p.333 8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices Problem Set p.338 8.4 Eigenbases. Diagonalization. Quadratic Forms Problem Set p.345 8.5 Complex Matrices and Forms. Problem Set p.351 Review Questions and Problems p.352

### Chapter 9

Vector Differential Calculus, Grad, Div, Curl

 9.1 Vectors in 2-Space and 3-Space Problem Set p.360 9.2 Inner Product (Dot Product) Problem Set p.367 9.3 Vector Product (Cross Product) Problem Set p.374 9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives Problem Set p.380 9.5 Curves. Arc Length. Curvature. Torsion Problem Set p.390 9.7 Gradient of a Scalar Field. Directional Derivative Problem Set p.402 9.8 Divergence of a Vector Field Problem Set p.405 9.9 Curl of a Vector Field Problem Set p.408 Review Questions and Problems p.409

### Chapter 10

Vector Integral Calculus. Integral Theorems

 10.1 Line Integrals Problem Set p.418 10.2 Path Independence of Line Integrals Problem Set p.425 10.3 Calculus Review: Double Integrals. Problem Set p.432 10.4 Green's Theorem in the Plane Problem Set p.438 10.5 Surfaces for Surface Integrals Problem Set p.442 10.6 Surface Integrals Problem Set p.450 10.7 Triple Integrals. Divergence Theorem of Gauss Problem Set p.457 10.8 Further Applications of the Divergence Theorem Problem Set p.462 10.9 Stokes's Theorem Problem Set p.468 Review Questions and Problems p.469

### Chapter 11

Fourier Analysis

 11.1 Fourier Series Problem Set p.482 11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions Problem Set p.490 11.3 Forced Oscillations Problem Set p.494 11.4 Approximation by Trigonometric Polynomials Problem Set p.498 11.5 Sturm-Liouville Problems. Orthogonal Functions Problem Set p.503 11.6 Orthogonal Series. Generalized Fourier Series Problem Set p.509 11.7 Fourier Integral Problem Set p.517 11.8 Fourier Cosine and Sine Transforms Problem Set p.522 11.9 Fourier Transform. Discrete and Fast Fourier Transforms Problem Set p.533 Review Questions and Problems p.537

### Chapter 12

Partial Differential Equations (Pdes)

 12.1 Basic Concepts of PDEs Problem Set p.542 12.3 Solution by Separating Variables. Use of Fourier Series Problem Set p.551 12.4 D'Alembert's Solution of the Wave Equation. Characteristics Problem Set p.556 12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem Problem Set p.566 12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms Problem Set p.574 12.9 Rectangular Membrane. Double Fourier Series Problem Set p.584 12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series Problem Set p.591 12.11 Laplace's Equation in Cylindrical and Spherical Coordinates. Potential Problem Set p.598 12.12 Solution of PDEs by Laplace Transforms Problem Set p.602 Review Questions and Problems p.603

### Chapter 13

Complex Numbers And Functions. Complex Differentiation

 13.1 Complex Numbers and Their Geometric Representation Problem Set p.612 13.2 Polar Form of Complex Numbers. Powers and Roots Problem Set p.618 13.3 Derivative. Analytic Function Problem Set p.624 13.4 Cauchy-Riemann Equations. Laplace's Equation Problem Set p.629 13.5 Exponential Function Problem Set p.632 13.6 Trigonometric and Hyperbolic Functions. Euler's Formula Problem Set p.636 13.7 Logarithm. General Power. Principal Value Problem Set p.640 Review Questions and Problems p.641

### Chapter 14

Complex Integration

 14.1 Line Integral in the Complex Plane Problem Set p.651 14.2 Cauchy's Integral Theorem Problem Set p.659 14.3 Cauchy's Integral Formula Problem Set p.663 14.4 Derivatives of Analytic Functions Problem Set p.667 Review Questions and Problems p.668

### Chapter 15

Power Series, Taylor Series

 15.1 Sequences, Series, Convergence Tests Problem Set p.679 15.2 Power Series Problem Set p.684 15.3 Functions Given by Power Series Problem Set p.689 15.4 Taylor and Maclaurin Series Problem Set p.697 15.5 Uniform Convergence. Problem Set p.704 Review Questions and Problems p.706

### Chapter 16

Laurent Series. Residue Integration

 16.1 Laurent Series Problem Set p.714 16.2 Singulariteis and Zeros. Infinity Problem Set p.719 16.3 Residue Integration Method Problem Set p.725 16.4 Residue Integration of Real Integrals Problem Set p.733 Review Questions and Problems p.733

### Chapter 17

Confomal Mapping

 17.1 Geometry of Analytic Functions: Conformal Mapping Problem Set p.741 17.2 Linear Fractional Transformations (Mobius Transformations) Problem Set p.745 17.3 Special Linear Fractional Transformations Problem Set p.750 17.4 Conformal Mapping by Other Functions Problem Set p.754 Review Questions and Problems p.756 17.5 Riemann Surfaces Problem Set p.756

### Chapter 18

Complex Analysis And Potential Theory

 18.1 Electrostactic Fields Problem Set p.762 18.2 Use of Conformal Mapping. Modeling Problem Set p.766 18.3 Heat Problems Problem Set p.769 18.4 Fluid Flow Problem Set p.776 18.5 Poisson's Integral Formula for Potentials Problem Set p.781 18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem Problem Set p.784 Review Questions and Problems p.785

### Chapter 19

Numerics In General

 19.1 Introduction Problem Set p.796 19.2 Solution of Equations by Iteration Problem Set p.807 19.3 Interpolation Problem Set p.819 19.4 Spline Interpolation Problem Set p.826 19.5 Numeric Integration and Differentiation Problem Set p.839 Review Questions and Problems p.841

### Chapter 20

Numeric Linear Algebra

 20.1 Linear Systems: Gauss Elimation Problem Set p.851 20.2 Linear Systems: LU-Factorization, Matrix Inversion Problem Set p.857 20.3 Linear Systems: Solution by Iteration Problem Set p.863 20.4 Linear Systems: Ill-Conditioning, Norms Problem Set p.871 20.5 Least Squares Method Problem Set p.875 20.7 Inclusion of Matrix Eigenvalues Problem Set p.884 20.8 Power Method for Eigenvalues Problem Set p.887 20.9 Tridiagonalization and QR-Factorization Problem Set p.896 Review Questions and Problems p.896

### Chapter 21

Numerics For Odes And Pdes

 21.1 Methods for First-Order ODEs Problem Set p.910 21.2 Multistep Methods Problem Set p.915 21.3 Methdos for Systems and Higher Order ODEs Problem Set p.922 21.4 Methods for Elliptic PDEs Problem Set p.930 21.5 Neumann and Mixed Problems. Irregular Boundary Problem Set p.935 21.6 Methods for Parabolic PDEs Problem Set p.941 21.7 Method for Hyberbolic PDEs Problem Set p.944 Review Questions and Problems p.945

### Chapter 22

Unconstrained Optimization. Linear Programming

 22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent Problem Set p.953 22.2 Linear Programming Problem Set p.957 22.3 Simplex Method Problem Set p.961 Review Questions and Problems p.968 22.4 Simplex Method: Difficulties Problem Set p.968

### Chapter 23

Graphs. Combinatorial Optimization

 23.1 Graphs and Digraphs Problem Set p.974 23.2 Shortest Path Problems. Complexity Problem Set p.979 23.3 Bellman's Principle. Dijkstra's Algorithm Problem Set p.983 23.4 Shortest Spanning Trees: Greedy Algorithm Problem Set p.987 23.5 Shortest Spanning Trees: Prims's Algorithm Problem Set p.990 23.6 Flows in Networks Problem Set p.997 23.7 Maximum Flow: Ford-Fulkerson Algorithm Problem Set p.1000 23.8 Bipartite Graphs. Assignment Problems Problem Set p.1005 Review Questions and Problems p.1006

### Chapter 24

Data Analysis, Probability Theory

 24.1 Data Representation. Average. Spread Problem Set p.1015 24.2 Experiments, Outcomes, Events Problem Set p.1017 24.3 Probability Problem Set p.1024 24.4 Permutations and Combinations Problem Set p.1028 24.5 Random Variables. Probability Distributions Problem Set p.1034 24.6 Mean and Variance of a Distribution Problem Set p.1038 24.7 Binomial, Poisson, and Hypergeometric Distributions Problem Set p.1044 24.8 Normal Distribution Problem Set p.1050 24.9 Distributions of Several Random Variables Problem Set p.1059 Review Questions and Problems p.1060

### Chapter 25

Mathematical Statistics

 25.2 Point Estimation of Parameters Problem Set p.1067 25.3 Confidence Intervals Problem Set p.1077 25.4 Testing of Hypotheses. Decisions Problem Set p.1086 25.5 Quality Control Problem Set p.1091 25.6 Acceptance Sampling Problem Set p.1095 25.7 Goodness of Fit. X^2-Test Problem Set p.1099 25.8 Nonparametric Tests Problem Set p.1102 25.9 Regression. Fitting Straight Lines. Correlation Problem Set p.1111 Review Questions and Problems p.1111