Advanced Engineering Mathematics SI Edition By Peter O Neil Solutions Manual

Contents 1 First-Order Differential Equations 1 1.1 Terminology and Separable Equations 1 1.2 The Linear First-Order Equation 12 1.3 Exact Equations 19 1.4 Homogeneous, Bernoulli and Riccati Equations 28 2 Second-O ...

Contents 1 First-Order Differential Equations 1 1.1 Terminology and Separable Equations 1 1.2 The Linear First-Order Equation 12 1.3 Exact Equations 19 1.4 Homogeneous, Bernoulli and Riccati Equations 28 2 Second-Order Differential Equations 37 2.1 The Linear Second-Order Equation 37 2.2 The Constant Coefficient Homogeneous Equation 41 2.3 Particular Solutions of the Nonhomogeneous Equation 46 2.4 The Euler Differential Equation 53 2.5 Series Solutions 58 3 The Laplace Transform 69 3.1 Definition and Notation 69 3.2 Solution of Initial Value Problems 72 3.3 The Heaviside Function and Shifting Theorems 77 3.4 Convolution 86 3.5 Impulses and the Dirac Delta Function 92 3.6 Systems of Linear Differential Equations 93 iii iv CONTENTS 4 Sturm-Liouville Problems and Eigenfunction Expansions 101 4.1 Eigenvalues and Eigenfunctions and Sturm-Liouville Problems 101 4.2 Eigenfunction Expansions 107 4.3 Fourier Series 114 5 The Heat Equation 137 5.1 Diffusion Problems on a Bounded Medium 137 5.2 The Heat Equation With a Forcing Term F(x, t) 147 5.3 The Heat Equation on the Real Line 150 5.4 The Heat Equation on a Half-Line 153 5.5 The Two-Dimensional Heat Equation 155 6 The Wave Equation 157 6.1 Wave Motion on a Bounded Interval 157 6.2 Wave Motion in an Unbounded Medium 167 6.3 d’Alembert’s Solution and Characteristics 173 6.4 The Wave Equation With a Forcing Term K(x, t) 190 6.5 The Wave Equation in Higher Dimensions 192 7 Laplace’s Equation 197 7.1 The Dirichlet Problem for a Rectangle 197 7.2 The Dirichlet Problem for a Disk 202 7.3 The Poisson Integral Formula 205 7.4 The Dirichlet Problem for Unbounded Regions 205 7.5 A Dirichlet Problem in 3 Dimensions 208 7.6 The Neumann Problem 211 7.7 Poisson’s Equation 217 8 Special Functions and Applications 221 8.1 Legendre Polynomials 221 8.2 Bessel Functions 235 8.3 Some Applications of Bessel Functions 251 9 Transform Methods of Solution 263 9.1 Laplace Transform Methods 263 9.2 Fourier Transform Methods 268 9.3 Fourier Sine and Cosine Transforms 271 10 Vectors and the Vector Space Rn 275 10.1 Vectors in the Plane and 3− Space 275 10.2 The Dot Product 277 10.3 The Cross Product 278 10.4 n− Vectors and the Algebraic Structure of Rn 280 10.5 Orthogonal Sets and Orthogonalization 284 10.6 Orthogonal Complements and Projections 287 11 Matrices, Determinants and Linear Systems 291 11.1 Matrices and Matrix Algebra 291 11.2. Row Operations and Reduced Matrices 295 11.3 Solution of Homogeneous Linear Systems 299 11.4 Nonhomogeneous Systems 306 11.5 Matrix Inverses 313 11.6 Determinants 315 11.7 Cramer’s Rule 318 11.8 The Matrix Tree Theorem 320 v 12 Eigenvalues, Diagonalization and Special Matrices 323 12.1 Eigenvalues and Eigenvectors 323 12.2 Diagonalization 327 12.3 Special Matrices and Their Eigenvalues and Eigenvectors 332 12.4 Quadratic Forms 336 13 Systems of Linear Differential Equations 339 13.1 Linear Systems 339 13.2 Solution of X0 = AX When A Is Constant 341 13.3 Exponential Matrix Solutions 348 13.4 Solution of X0 = AX + G for Constant A 350 13.5 Solution by Diagonalization 353 14 Nonlinear Systems and Qualitative Analysis 359 14.1 Nonlinear Systems and Phase Portraits 359 14.2 Critical Points and Stability 363 14.3 Almost Linear Systems 364 14.4 Linearization 369 15 Vector Differential Calculus 373 15.1 Vector Functions of One Variable 373 15.2 Velocity, Acceleration and Curvature 376 15.3 The Gradient Field 381 15.4 Divergence and Curl 385 15.5 Streamlines of a Vector Field 387 16 Vector Integral Calculus 391 16.1 Line Integrals 391 16.2 Green’s Theorem 393 16.3 Independence of Path and Potential Theory 398 16.4 Surface Integrals 405 16.5 Applications of Surface Integrals 408 16.6 Gauss’s Divergence Theorem 412 16.7 Stokes’s Theorem 414 17 Fourier Series 419 17.1 Fourier Series on [−L, L] 419 17.2 Sine and Cosine Series 423 17.3 Integration and Differentiation of Fourier Series 428 17.4 Properties of Fourier Coefficients 430 17.5 Phase Angle Form 432 17.6 Complex Fourier Series 435 17.7 Filtering of Signals 438 vi CONTENTS 18 Fourier Transforms 441 18.1 The Fourier Transform 441 18.2 Fourier sine and Cosine Transforms 448 19 Complex Numbers and Functions 451 19.1 Geometry and Arithmetic of Complex Numbers 451 19.2 Complex Functions 455 19.3 The Exponential and Trigonometric Functions 461 19.4 The Complex Logarithm 467 19.5 Powers 468 20 Complex Integration 473 20.1 The Integral of a Complex Function 473 20.2 Cauchy’s Theorem 477 20.3 Consequences of Cauchy’s Theorem 479 21 Series Representations of Functions 485 21.1 Power Series 485 21.2 The Laurent Expansion 492 22 Singularities and the Residue Theorem 497 22.1 Classification of Singularities 497 22.2 The Residue Theorem 499 22.3 Evaluation of Real Integrals 505 23 Conformal Mappings 515 23.1 The Idea of a Conformal Mapping 515 23.2 Construction of Conformal Mappings 533