A Student’s Solutions Manual to Accompany
ADVANCED ENGINEERING MATHEMATICS,
8TH EDITION
PETER V. O’NEIL
Elitestudyvault
,Contents
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
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,iv 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 179
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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 248
Elitestudyvault
ADVANCED ENGINEERING MATHEMATICS,
8TH EDITION
PETER V. O’NEIL
Elitestudyvault
,Contents
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
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,iv 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 179
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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 248
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