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

iii

, iv CONTENTS

4 Sturm-Liouville Problems and Eigenfunction Expansions 53
4.1 Eigenvalues
cd and Eigenfunctionsc d c d and Sturm-c d c d 53 c d c d

Liouville Problems c d c d

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


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

Term t)
5.3 The Heat Equation on the Real Lin
cd c d 79
c d c d c d c d

e
5.4 The Heat Equation on a Half-Line
cd cdcd c 81 d cdcd cdcd

5.5 The Two-Dimensional
cd Heat Equation 82
6 The Wave Equation
c d c d c d c d 85
6.1 Wave Motion on a Bounded Interval
cd c d c d c d85 c d c d


6.2 Wave Motion in an Unbounded Medium
cd c d c d c d 90 c d c d

6.3 d’Alembert’s Solution
cd and Characteristics
c d c d 95 c d c d c d c d


6.4 The cd Wave Equation
c d c d With a Forcing
c d c d Te t) 103 c d c d c d c d c d c d c d c d

rm K(x,
c d c d

6.5 The cd Wave Equation
c d c d in Higher Dimensions
c d c d 105 c d c d c d c d c d c d

7 Laplace’s Equation c d c d 107
7.1 The cd Dirichlet Problem
c d c d for a Rectangle
c d c d 107 c d c d c d c d c d c d


7.2 The cd Dirichlet Problem
c d c d for a Disk c d c d 110 c d c d c d c d c d c d

7.3 The Poisson Integral Formula
cd cdcd cdcd 112 cdcd


7.4 The cd Dirichlet Problem
c d c d for Unbounded Reg
c d c d 112 c d c d c d c d c d c d

ions
7.5 A cd Dirichlet
c d c dProblem in 3 Dimensions
c d c d 114 c d c d c d c d c d c d


7.6 The cd Neumann c d c dProblem c d c d 115
7.7 Poisson’s
cd Equation c d c d 119
8 Special Functions and
c d c d Applications 121
c d c d c d c d

8.1 Legendre
cd Polynomials c d c d 121
8.2 Bessel
cd Functions c d c d 129
8.3 Some Applications of Bessel Functions
cd cd c d 138
c d c d c d c d c d c d

9 Transform Methods of Solution 145
9.1 Laplace Transform Methods
cd 145
9.2 Fourier Transform
cd Methods
c d c d 148
c d c d

9.3 Fourier Sine and Cosine Transforms
cd c d c d c d c d 150
c d c d c d c d


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

Rn
c d c d

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

10.2 The Dot Product —
c d c d c d c d 154
10.3 The cdCross Product
c d c d c d c d 155
10.4 n of Rn 156
— and the Algebraic Structure
cd

Vectors c d c d c d c d
cd
10.5 Orthogonal Sets and Orthogzation
cd c d c d 158
c d c d c d c d

onali
10.6 Orthogonal Complements androjections
cd cdcd 160 cdcd

P
cdcd

11 Matrices, Determinants a Linear
c d c d Systems 163 c d c d c d c d

nd
11.1 Matricescd and Matrix Alge
c d c d c d c d 163 c d c d

bra
11.2. Row Operations and Red Matrices
c d c d c d 165
c d c d c d c d

uced
11.3 Solution of Homogeneous Li Systems
cd c d c d 167 c d

near
11.4 Nonhomogeneous Systems
cd c d 171