ADVANCED ENGINEERING MATHEMATICS SI EDITION 8TH EDITION BY PETER O NEIL, SOLUTIONS MANUAL TEST BANK .

A jStudent’s jSolutions jManual jto jAccompany
ADVANCED ENGINEERING
j


MATHEMATICS,8TH EDITION
j j j



PETER jV. jO’NEIL

, STUDENT'S SOLUTIONS MANUAL
j j




TO ACCOMPANY
j




Advanced
EngineeringMathematics
j j




8th EDITION
j




PETER jV. jO’NEIL

,Contents




1 First-Order j Differential j Equations 1
1.1 Terminology j and j Separable j Equations 1
1.2 The j Linear j First-Order j Equation 8
1.3 Exact j Equations 11
1.4 Homogeneous, j Bernoulli j and j Riccati j Equations 15
2 Second-Order j Differential j Equations 19
2.1 The j Linear j Second-Order j Equation 19
2.2 The j Constant j Coefficient j Homogeneous j Equation 21
2.3 Particular j Solutions j of j the j Nonhomogeneous j Equation 24
2.4 The j Euler j Differential j Equation 27
2.5 Series j Solutions 29
3 The j Laplace j Transform 35
3.1 Definition j and j Notation 35
3.2 Solution j of j Initial j Value j Problems 37
3.3 The j Heaviside j Function j and j Shifting j Theorems 40
3.4 Convolution 44
3.5 Impulses j and j the j Dirac j Delta j Function 48
3.6 Systems j of j Linear j Differential j Equations 48
iii

, iv CONTENTS

4 Sturm-Liouville j Problems j and j Eigenfunction j Expansions 53
4.1 Eigenvalues j and j Eigenfunctions j and j Sturm-Liouville j Problems 53
4.2 Eigenfunction j j Expansions 57
4.3 Fourier j Series 61
5 The j Heat j Equation 71
5.1 Diffusion j Problems j on j a j Bounded j Medium 71
5.2 The j Heat j Equation j With j a j Forcing j Term j F j(x, jt) 76
5.3 The j Heat j Equation j on j the j Real j Line 79
5.4 The j Heat j Equation j on j a j Half-Line 81
5.5 The j jTwo-Dimensional j jHeat j jEquation 82
6 The j Wave j Equation 85
6.1 Wave j Motion j on j a j Bounded j Interval 85
6.2 Wave j Motion j in j an j Unbounded j Medium 90
6.3 d’Alembert’s j Solution j and j Characteristics 95
6.4 The j Wave j Equation j With j a j Forcing j Term j K(x, jt) 103
6.5 The j Wave j Equation j in j Higher j Dimensions 105
7 Laplace’s j Equation 107
7.1 The j Dirichlet j Problem j for j a j Rectangle 107
7.2 The j Dirichlet j Problem j for j a j Disk 110
7.3 The jPoisson jIntegral jFormula 112
7.4 The j Dirichlet j Problem j for j Unbounded j Regions 112
7.5 A j Dirichlet j Problem j in j 3 j Dimensions 114
7.6 The j Neumann j Problem 115
7.7 Poisson’s j Equation 119
8 Special j Functions j and j Applications 121
8.1 Legendre j Polynomials 121
8.2 Bessel j Functions 129
8.3 Some j Applications j of j Bessel j Functions 138
9 Transform j Methods j of j Solution 145
9.1 Laplace j Transform j Methods 145
9.2 Fourier j Transform j Methods 148
9.3 Fourier j Sine j and j Cosine j Transforms 150
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10 Vectors j and j the j Vector j Space j R 153
10.1 Vectors j in j the j Plane j and−j 3 Space 153
10.2 The j Dot j Product 154
10.3 The j Cross j Product 155
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10.4 n− Vectors j and j the j Algebraic j Structure j of j R 156
10.5 Orthogonal j Sets j and j Orthogonalization 158
10.6 Orthogonal j Complements j and j Projections 160
11 Matrices, j Determinants j and j Linear j Systems 163
11.1 Matrices j and j Matrix j Algebra 163
11.2. j Row j Operations j and j Reduced j Matrices 165
11.3 Solution j of j Homogeneous j Linear j Systems 167
11.4 Nonhomogeneous j Systems 171
11.5 Matrix j Inverses 175
11.6 Determinants 176
11.7 Cramer’s j j Rule 178
11.8 The j Matrix j Tree j Theorem 179