Chapter 1
First-Order Differential
Equations
1.1 Terminology and Separable Equations
1. The differential equation is separable because it can be written
3y
2
dy
= 4x,
dx
or, in differential form,
Integrate to obtain
3y
2 dy = 4xdx.
y
3
= 2x
2
+ k.
This implicitly defines a genera...
Solution Manual for Advanced Engineering
Mathematics 8th Edition O’Neil
PETER V. O’NEIL
,Contents
1 First-Order Differential Equations 1
1.1 Terminology and Separable Equations 1
1.2 The Linear First-Order Equation 12
1.3 Exact Equations 19
1.4 Homogeneous, Bernoulli and Riccati Equations 28
2 Second-Order Differential Equations 37
2.1 The Linear Second-Order Equation 37
2.2 The Constant Coefficient Homogeneous Equation 41
2.3 Particular Solutions of the Nonhomogeneous Equation 46
2.4 The Euler Differential Equation 53
2.5 Series Solutions 58
3 The Laplace Transform 69
3.1 Definition and Notation 69
3.2 Solution of Initial Value Problems 72
3.3 The Heaviside Function and Shifting Theorems 77
3.4 Convolution 86
3.5 Impulses and the Dirac Delta Function 92
3.6 Systems of Linear Differential Equations 93
iii
,iv CONTENTS
4 Sturm-Liouville Problems and Eigenfunction Expansions 101
4.1 Eigenvalues and Eigenfunctions and Sturm-Liouville Problems 101
4.2 Eigenfunction Expansions 107
4.3 Fourier Series 114
5 The Heat Equation 137
5.1 Diffusion Problems on a Bounded Medium 137
5.2 The Heat Equation With a Forcing Term F (x, t) 147
5.3 The Heat Equation on the Real Line 150
5.4 The Heat Equation on a Half-Line 153
5.5 The Two-Dimensional Heat Equation 155
6 The Wave Equation 157
6.1 Wave Motion on a Bounded Interval 157
6.2 Wave Motion in an Unbounded Medium 167
6.3 d’Alembert’s Solution and Characteristics 173
6.4 The Wave Equation With a Forcing Term K(x, t) 190
6.5 The Wave Equation in Higher Dimensions 192
7 Laplace’s Equation 197
7.1 The Dirichlet Problem for a Rectangle 197
7.2 The Dirichlet Problem for a Disk 202
7.3 The Poisson Integral Formula 205
7.4 The Dirichlet Problem for Unbounded Regions 205
7.5 A Dirichlet Problem in 3 Dimensions 208
7.6 The Neumann Problem 211
7.7 Poisson’s Equation 217
8 Special Functions and Applications 221
8.1 Legendre Polynomials 221
8.2 Bessel Functions 235
8.3 Some Applications of Bessel Functions 251
9 Transform Methods of Solution 263
9.1 Laplace Transform Methods 263
9.2 Fourier Transform Methods 268
9.3 Fourier Sine and Cosine Transforms 271
10 Vectors and the Vector Space Rn 275
10.1 Vectors in the Plane and 3 − Space 275
10.2 The Dot Product 277
10.3 The Cross Product 278
10.4 n− Vectors and the Algebraic Structure of Rn 280
10.5 Orthogonal Sets and Orthogonalization 284
10.6 Orthogonal Complements and Projections 287
11 Matrices, Determinants and Linear Systems 291
11.1 Matrices and Matrix Algebra 291
11.2. Row Operations and Reduced Matrices 295
11.3 Solution of Homogeneous Linear Systems 299
11.4 Nonhomogeneous Systems 306
11.5 Matrix Inverses 313
11.6 Determinants 315
11.7 Cramer’s Rule 318
11.8 The Matrix Tree Theorem 320
, v
12 Eigenvalues, Diagonalization and Special Matrices 323
12.1 Eigenvalues and Eigenvectors 323
12.2 Diagonalization 327
12.3 Special Matrices and Their Eigenvalues and Eigenvectors 332
12.4 Quadratic Forms 336
13 Systems of Linear Differential Equations 339
13.1 Linear Systems 339
13.2 Solution of X′ = AX When A Is Constant 341
13.3 Exponential Matrix Solutions 348
13.4 Solution of X′ = AX + G for Constant A 350
13.5 Solution by Diagonalization 353
14 Nonlinear Systems and Qualitative Analysis 359
14.1 Nonlinear Systems and Phase Portraits 359
14.2 Critical Points and Stability 363
14.3 Almost Linear Systems 364
14.4 Linearization 369
15 Vector Differential Calculus 373
15.1 Vector Functions of One Variable 373
15.2 Velocity, Acceleration and Curvature 376
15.3 The Gradient Field 381
15.4 Divergence and Curl 385
15.5 Streamlines of a Vector Field 387
16 Vector Integral Calculus 391
16.1 Line Integrals 391
16.2 Green’s Theorem 393
16.3 Independence of Path and Potential Theory 398
16.4 Surface Integrals 405
16.5 Applications of Surface Integrals 408
16.6 Gauss’s Divergence Theorem 412
16.7 Stokes’s Theorem 414
17 Fourier Series 419
17.1 Fourier Series on [−L, L] 419
17.2 Sine and Cosine Series 423
17.3 Integration and Differentiation of Fourier Series 428
17.4 Properties of Fourier Coefficients 430
17.5 Phase Angle Form 432
17.6 Complex Fourier Series 435
17.7 Filtering of Signals 438
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