INSTRUCTORS SOLUTIONS MANUAL

INSTRUCTOR'S SOLUTIONS MANUAL

TO ACCOMPANY

ADVANCED ENGINEERNG

MATHEMATICS

8 th

EDITION

PETER V. O’NEIL 1 / 4

Contents

• First-Order Dierential 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

• Second-Order Dierential Equations 37

2.1 The Linear Second-Order Equation 37 2.2 The Constant Coecient Homogeneous Equation 41 2.3 Particular Solutions of the Nonhomogeneous Equation 46 2.4 The Euler Dierential Equation 53 2.5 Series Solutions 58

• The Laplace Transform 69

3.1 Denition 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 Dierential Equations 93 iii 2 / 4

ivCONTENTS

• 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

• The Heat Equation 137

5.1 Diusion Problems on a Bounded Medium 137 5.2 The Heat Equation With a Forcing TermF(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

• 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 TermK(x; t) 190 6.5 The Wave Equation in Higher Dimensions 192

• 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

• Special Functions and Applications 221

8.1 Legendre Polynomials 221 8.2 Bessel Functions 235 8.3 Some Applications of Bessel Functions 251

• 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 R n 275 10.1 Vectors in the Plane and 3Space 275 10.2 The Dot Product 277 10.3 The Cross Product 278 10.4nVectors and the Algebraic Structure ofR n 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 3 / 4

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 Dierential Equations 339 13.1 Linear Systems 339 13.2 Solution ofX

=AXWhenAIs Constant 341 13.3 Exponential Matrix Solutions 348 13.4 Solution ofX

=AX+Gfor ConstantA350 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 Dierential 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 Dierentiation of Fourier Series 428 17.4 Properties of Fourier Coecients 430 17.5 Phase Angle Form 432 17.6 Complex Fourier Series 435 17.7 Filtering of Signals 438

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