A Student’s Solutions Manual to Accompany ADVANCED ENGINEERING MATHEMATICS, 8TH EDITION PETER V. O’NEIL

STUDENT'S SOLUTIONS MANUAL TO ACCOMPANY Advanced Engineering Mathematics 8th EDITION PETER V. O’NEILContents 1 First-Order Differential Equations 1 1.1 Terminology and Separable Equations 1 1.2 The Linear First-Order Equation 8 1.3 Exact Equations 11 1.4 Homogeneous, Bernoulli and Riccati Equations 15 2 Second-Order Differential Equations 19 2.1 The Linear Second-Order Equation 19 2.2 The Constant Coefficient Homogeneous Equation 21 2.3 Particular Solutions of the Nonhomogeneous Equation 24 2.4 The Euler Differential Equation 27 2.5 Series Solutions 29 3 The Laplace Transform 35 3.1 Definition and Notation 35 3.2 Solution of Initial Value Problems 37 3.3 The Heaviside Function and Shifting Theorems 40 3.4 Convolution 44 3.5 Impulses and the Dirac Delta Function 48 3.6 Systems of Linear Differential Equations 48 iiiiv CONTENTS 4 Sturm-Liouville Problems and Eigenfunction Expansions 53 4.1 Eigenvalues and Eigenfunctions and Sturm-Liouville Problems 53 4.2 Eigenfunction Expansions 57 4.3 Fourier Series 61 5 The Heat Equation 71 5.1 Diffusion Problems on a Bounded Medium 71 5.2 The Heat Equation With a Forcing Term F (x; t) 76 5.3 The Heat Equation on the Real Line 79 5.4 The Heat Equation on a Half-Line 81 5.5 The Two-Dimensional Heat Equation 82 6 The Wave Equation 85 6.1 Wave Motion on a Bounded Interval 85 6.2 Wave Motion in an Unbounded Medium 90 6.3 d’Alembert’s Solution and Characteristics 95 6.4 The Wave Equation With a Forcing Term K(x; t) 103 6.5 The Wave Equation in Higher Dimensions 105 7 Laplace’s Equation 107 7.1 The Dirichlet Problem for a Rectangle 107 7.2 The Dirichlet Problem for a Disk 110 7.3 The Poisson Integral Formula 112 7.4 The Dirichlet Problem for Unbounded Regions 112 7.5 A Dirichlet Problem in 3 Dimensions 114 7.6 The Neumann Problem 115 7.7 Poisson’s Equation 119 8 Special Functions and Applications 121 8.1 Legendre Polynomials 121 8.2 Bessel Functions 129 8.3 Some Applications of Bessel Functions 138 9 Transform Methods of Solution 145 9.1 Laplace Transform Methods 145 9.2 Fourier Transform Methods 148 9.3 Fourier Sine and Cosine Transforms 150 10 Vectors and the Vector Space Rn 153 10.1 Vectors in the Plane and 3− Space 153 10.2 The Dot Product 154 10.3 The Cross Product 155 10.4 n− Vectors and the Algebraic Structure of Rn 156 10.5 Orthogonal Sets and Orthogonalization 158 10.6 Orthogonal Complements and Projections 160 11 Matrices, Determinants and Linear Systems 163 11.1 Matrices and Matrix Algebra 163 11.2. Row Operations and Reduced Matrices 165 11.3 Solution of Homogeneous Linear Systems 167 11.4 Nonhomogeneous Systems 171 11.5 Matrix Inverses 175 11.6 Determinants 176 11.7 Cramer’s Rule 178 11.8 The Matrix Tree Theorem 179v 12 Eigenvalues, Diagonalization and Special Matrices 181 12.1 Eigenvalues and Eigenvectors 181 12.2 Diagonalization 183 12.3 Special Matrices and Their Eigenvalues and Eigenvectors 186 12.4 Quadratic Forms 188 13 Systems of Linear Differential Equations 189 13.1 Linear Systems 189 13.2 Solution of X0 = AX When A Is Constant 190 13.3 Exponential Matrix Solutions 194 13.4 Solution of X0 = AX + G for Constant A 195 13.5 Solution by Diagonalization 197 14 Nonlinear Systems and Qualitative Analysis 201 14.1 Nonlinear Systems and Phase Portraits 201 14.2 Critical Points and Stability 204 14.3 Almost Linear Systems 205 14.4 Linearization 209 15 Vector Differential Calculus 211 15.1 Vector Functions of One Variable 211 15.2 Velocity, Acceleration and Curvature 213 15.3 The Gradient Field 216 15.4 Divergence and Curl 218 15.5 Streamlines of a Vector Field 219 16 Vector Integral Calculus 223 16.1 Line Integrals 223 16.2 Green’s Theorem 224 16.3 Independence of Path and Potential Theory 226 16.4 Surface Integrals 229 16.5 Applications of Surface Integrals 230 16.6 Gauss’s Divergence Theorem 233 16.7 Stokes’s Theorem 234 17 Fourier Series 237 17.1 Fourier Series on [−L; L] 237 17.2 Sine and Cosine Series 239 17.3 Integration and Differentiation of Fourier Series 242 17.4 Properties of Fourier Coefficients 244 17.5 Phase Angle Form 245 17.6 Complex Fourier Series 247 17.7 Filtering of Signals 248vi CONTENTS 18 Fourier Transforms 251 18.1 The Fourier Transform 251 18.2 Fourier sine and Cosine Transforms 254 19 Complex Numbers and Functions 257 19.1 Geometry and Arithmetic of Complex Numbers 257 19.2 Complex Functions 259 19.3 The Exponential and Trigonometric Functions 262 19.4 The Complex Logarithm 264 19.5 Powers 265 20 Complex Integration 269 20.1 The Integral of a Complex Function 269 20.2 Cauchy’s Theorem 271 20.3 Consequences of Cauchy’s Theorem 272 21 Series Representations of Functions 275 21.1 Power Series 275 21.2 The Laurent Expansion 279 22 Singularities and the Residue