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

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

, STUDENT'S SOLUTIONS MANUAL
TO ACCOMPANY




AdvancedEngineering
Mathematics
8th EDITION




PETER V. O’NEIL

,Contents




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

iii

, iv CONTENTS

4 Sturm-Liouville Problems and Eigenfunction Expansions 53
4.1 Eigenvalues
xc and Eigenfunctions and
x c x c Sturm- 53
x c x c x c x c

Liouville Problems x c x c

4.2 Eigenfunction Expansions
xc 57
4.3 Fourier
xc Series x c x c 61
5 The Heat Equation 71
5.1 Diffusion
xc Problems on a Bounded
x c x c Medium 71 x c x c x c x c x c x c x c x c


5.2 The Heat Equation With a Forcing
xc x c F (x, 76 x c x c x c x c x c x c

Term t)
5.3 The Heat Equation on the Real Line
xc x c 79 x c x c x c x c

5.4 The Heat Equation on a Half-Line
xc xcxc 81 x c xcxc xcxc

5.5 The Two-Dimensional
xc Heat Equation 82
6 The Wave Equation
x c x c x c x c 85
6.1 Wave Motion on a Bounded Interval
xc x c x c 85 x c x c x c


6.2 Wave Motion in an Unbounded Medium
xc x c x c 90 x c x c x c

6.3 d’Alembert’s Solution and Characteristics
xc x c x c 95 x c x c x c x c


6.4 The Wave Equation With a Forcing Ter t)
xc x c x c x c x c 103 x c x c x c x c x c x c x c x c

m K(x,x c x c

6.5 The xc Wave Equation
x c x c in Higher Dimensions
x c x c 105 x c x c x c x c x c x c

7 Laplace’s Equation x c x c 107
7.1 The xc Dirichlet Problem
x c x c for a Rectangle x c x c 107 x c x c x c x c x c x c

7.2 The xc Dirichlet Problem
x c x c for a Disk x c x c 110 x c x c x c x c x c x c


7.3 The Poisson Integral Formula
xc xcxc xcxc 112 xcxc

7.4 The xc Dirichlet Problem
x c x c for Unbounded Regi
x c x c 112 x c x c x c x c x c x c

ons
7.5 A Dirichlet Problem in 3 Dimensions
xc x c x c x c x c 114 x c x c x c x c x c x c

7.6 The Neumann
xc Problem
x c x c x c x c 115
7.7 Poisson’s
xc Equation x c x c 119
8 Special Functions and
x c x c Applications 121
x c x c x c x c

8.1 Legendre Polynomials
xc x c x c 121
8.2 Bessel Functions
xc x c x c 129
8.3 Some Applications of Bessel Functions
xc xc xc 138 x c x c x c xc xc x c

9 Transform Methods of Solution 145
9.1 Laplace Transform Methods
xc 145
9.2 Fourier Transform Methods
xc x c x c 148 x c x c

9.3 Fourier Sine and Cosine Transforms
xc x c x c x c x c 150 x c x c x c x c


10 Vectors and the Vector
x c x c Space x c x c 153 x c x c x c x c x

Rnc x c

10.1 Vectors in the Plane and 3 x c x c x c x c
Space153
x c x c x c x c x c x c

10.2 The Dot Product —
x c x c x c x c 154
10.3 The Cross Product
xc x c x c x c x c 155
10.4 n of Rn 156
— and the Algebraic Structure
xc

Vectors x c x c x c x c
xc
10.5 Orthogonal Sets and Orthog zation
xc x c x c 158
x c x c x c x c

onali
10.6 Orthogonal Complements and rojections
xc xcxc 160 xcxc

P
xcxc

11 Matrices, Determinants anLinear
x c x c Systems 163 x c x c x c x c

d
11.1 Matricesxc and Matrix Algeb x c x c x c x c163 x c x c

ra
11.2. Row Operations and Redu Matrices
x c x c x c 165 x c x c x c x c

ced
11.3 Solution of Homogeneous Li Systems
xc x c x c 167 x c

near
11.4 Nonhomogeneous Systems
xc 171
x c

11.5 Matrix Inverses
xc 175