I NSTRUCTOR ’ S S OLUTIONS M ANUAL TO ACCOMPANY
ADVANCED ENGINEERING MATHEMATICS SEVENTH EDITION
PETER V. O’NEIL
© 2012 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.
© 2012 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.
Contents 1
First-Order Differential Equations 1.1 Terminology and Separable Equations 1.2 Linear Equations 1.3 Exact Equations 1.4 Homogeneous, Bernoulli and Riccati Equations 1.5 Additional Applications 1.6 Existence and Uniqueness Questions 2 Linear Second-Order Equations 2.1 Theory of the Linear Second-Order Equation 2.2 The Constant Coefficient Case 2.3 The Nonhomogeneous Equation 2.4 Spring Motion 2.5 Euler’s Equation
1 1 14 19 26 30 39 42 42 45 48 53 63
3
The Laplace Transform 3.1 Definition and Notation 3.2 Solution of Initial Value Problems 3.3 Shifting and the Heaviside Function 3.4 Convolution 3.5 Impulses and the Delta Function 3.6 Solution of Systems 3.7 Polynomial Coefficients
66 66 70 73 81 89 91 100
Series Solutions 4.1 Power Series Solutions 4.2 Frobenius Solutions 5 Approximation of Solutions 5.1 Direction Fields 5.2 Euler’s Method 5.3 Taylor and Modified Euler Methods
103 103 107 112 112 115 118
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121 121 122 123 125 129 131 133
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Vectors and Vector Spaces 6.1 Vectors in the Plane and 3-Space 6.2 The Dot Product 6.3 The Cross Product 6.4 The Vector Space Rn 6.5 Orthogonalization 6.6 Orthogonal Complements and Projections 6.7 The Function Space C[a, b]
iii © 2012 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.
CONTENTS
iv 7
Matrices and Systems of Linear Equations 7.1 Matrices 7.2 Elementary Row Operations 7.3 Reduced Row Echelon Form 7.4 Row and Column Spaces 7.5 Homogeneous Systems 7.6 Nonhomogeneous Systems 7.7 Matrix Inverses 7.8 Least Squares Vectors and Data Fitting 7.9 LU Factorization 7.10 Linear Transformations Determinants 8.1 Definition of the Determinant 8.2 Evaluation of Determinants I 8.3 Evaluation of Determinants II 8.4 A Determinant Formula for A−1 8.5 Cramer’s Rule 8.6 The Matrix Tree Theorem
138 138 142 145 147 149 155 162 164 168 172 174 174 175 176 178 179 180
Eigenvalues and Diagonalization 9.1 Eigenvalues and Eigenvectors 9.2 Diagonalization 9.3 Some Special Types of Matrices 10 Systems of Linear Differential Equations 10.1 Linear Systems 10.2 Solution of X = AX for Constant A 10.3 Solution of X = AX + G 10.4 Exponential Matrix Solutions 10.5 Applications and Illustrations of Techniques 10.6 Phase Portraits 11 Vector Differential Calculus 11.1 Vector Functions of One Variable 11.2 Velocity and Curvature 11.3 Vector Fields and Streamlines 11.4 The Gradient Field 11.5 Divergence and Curl 12 Vector Integral Calculus 12.1 Line Integrals 12.2 Green’s Theorem 12.3 An Extension of Green’s Theorem 12.4 Independence of Path and Potential Theory 12.5 Surface Integrals 12.6 Applications of Surface Integrals 12.7 Lifting Green’s Theorem to R3 12.8 The Divergence Theorem of Gauss 12.9 Stokes’s Theorem 12.10 Curvilinear Coordinates
182 182 187 193 200 200 202 208 216 219 228 238 238 241 246 248 252 255 255 257 260 262 268 270 274 274 276 279
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9
© 2012 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.
CONTENTS 13
v
Fourier Series 13.1 Why Fourier Series? 13.2 The Fourier Series of a Function 13.3 Sine and Cosine Series 13.4 Integration and Differentiation of Fourier Series 13.5 Phase Angle Form 13.6 Complex Fourier Series 13.7 Filtering of Signals The Fourier Integral and Transforms 14.1 The Fourier Integral 14.2 Fourier Cosine and Sine Integrals 14.3 The Fourier Transform 14.4 Fourier Cosine and Sine Transforms 14.5 The Discrete Fourier Transform 14.6 Sampled Fourier Series 14.7 DFT Approximation of the Fourier Transform Special Functions and Eigenfunction Expansions 15.1 Eigenfunction Expansions 15.2 Legendre Polynomials 15.3 Bessel Functions
283 283 284 293 306 309 311 314 327 327 331 335 346 347 353 358 360 360 371 380
The Wave Equation 16.1 Derivation of the Wave Equation 16.2 Wave Motion on an Interval 16.3 Wave Motion in an Infinite Medium 16.4 Wave Motion in a Semi-Infinite Medium 16.5 Laplace Transform Techniques 16.6 Characteristics and d’Alembert’s Solution 16.7 Vibrations in a Circular Membrane I 16.8 Vibrations in a Circular Membrane II 16.9 Vibrations in a Rectangular Membrane
402 402 403 421 426 429 432 443 449 450
The Heat Equation 17.1 Initial and Boundary Conditions 17.2 The Heat Equation on [0, L] 17.3 Solutions in an Infinite Medium 17.4 Laplace Transform Techniques 17.5 Heat Conduction in an Infinite Cylinder 17.6 Heat Conduction in a Rectangular Plate 18 The Potential Equation 18.1 Laplace’s Equation 18.2 Dirichlet Problem for a Rectangle 18.3 Dirichlet Problem for a Disk 18.4 Poisson’s Integral Formula 18.5 Dirichlet Problem for Unbounded Regions 18.6 A Dirichlet Problem for a Cube 18.7 Steady-State Equation for a Sphere 18.8 The Neumann Problem
453 453 454 478 484 487 488 491 491 492 497 501 502 505 508 510
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15
16
17
© 2012 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part.