Differential equations

Preface ......Page 4
Contents ......Page 6
PART ONE - FUNDAMENTAL METHODS AND APPLICATIONS ......Page 9
1.1 Classification of Differential Equations ......Page 11
1.2 Solutions ......Page 15
1.3 Initial-Value problems, Boundary-Value problems, and Existence of solutions ......Page 23
2.1 Exact Differential Equations and Integrating Factors ......Page 33
2.2 Separable Equations and Equations Reducible to This Form ......Page 47
2.3 Linear Equations and Bernoulli Equations ......Page 57
2.4 Special Integrating Factors and Transformations ......Page 69
3.1 Orthogonal and Oblique Trajectories ......Page 78
3.2 Problems in Mechanics ......Page 85
3.3 Rate Problems ......Page 97
4.1 Basic Theory of Linear Differential Equations ......Page 110
4.2 The Homogeneous Linear Equation with Constant Coefficients ......Page 133
4.3 The Method of Undetermined Coefficients ......Page 145
4.4 Variation of Parameters ......Page 163
4.5 The Cauchy-Euler Equation ......Page 172
4.6 Statements and Proofs of Theorems on the Second-Order Homogeneous Linear \r\nEquation ......Page 178
5.1 The Differential Equation of the Vibrations of a Mass on a Spring ......Page 187
5.2 Free, Undamped Motion ......Page 190
5.3 Free, Damped Motion ......Page 197
5.4 Forced Motion ......Page 207
5.5 Resonance Phenomena ......Page 214
5.6 Electric Circuit Problems ......Page 219
6.1 Power Series Solutions About an Ordinary Point ......Page 229
6.2 Solutions About Singular Points; The Method of Frobenius ......Page 241
6.3 Bessel's Equation and Bessel Functions ......Page 260
7.1 Differential Operators and an Operator Method ......Page 272
7.2 Applications ......Page 286
7.3 Basic Theory of Linear Systems in Normal Form ......Page 298
7.4 Homogeneous Linear Systems with Constant Coefficients ......Page 309
7.5 Matrices and Vectors ......Page 320
7.6 The Matrix Method for Homogeneous Linear Systems with Constant Coefficients ......Page 354
7.7 The Matrix Method for Homogeneous Linear Systems with Constant Coefficients ......Page 363
8.1 Graphical Methods ......Page 385
8.2 Power Series Methods ......Page 392
8.3 The Method of Successive Approximations ......Page 398
8.4 Numerical Methods ......Page 402
9.1 Definition, Existence, and Basic Properties of the Laplace Transform ......Page 419
9.2 The Inverse Transform and the Convolution ......Page 439
9.3 Laplace Transform Solution of Linear Differential Equations with Constant Coefficients ......Page 449
9.4 Laplace Transform Solution of Linear Systems ......Page 461
PART TWO - FUNDAMENTAL THEORY AND FURTHER METHODS ......Page 466
10.1 Some Concepts from Real Function Theory ......Page 469
10.2 The Fundamental Existence and Uniqueness Theorem ......Page 481
10.3 Dependence of Solutions on Initial Conditions and on the Function ......Page 496
10.4 Existence and Uniqueness Theorems for Systems and Higher-Order Equations ......Page 503
11.1 Introduction ......Page 513
11.2 Basic Theory of the Homogeneous Linear System ......Page 518
11.3 Further Theory of the Homogeneous Linear System ......Page 530
11.4 The Nonhomogeneous Linear System ......Page 541
11.5 Basic Theory of the nth-Order Homogeneous Linear Differential Equation ......Page 551
11.6 Further Properties of the nth-Order Homogeneous Linear Differential Equation ......Page 566
11.7 The nth-Order Nonhomogeneous Linear Equation ......Page 577
11.8 Sturm Theory ......Page 581
12.1 Sturm-Liouville Problems ......Page 596
12.2 Orthogonality of Characteristic Functions ......Page 605
12.3 The Expansion of a Function in a Series of Orthonormal Functions ......Page 609
12.4 Trigonometric Fourier Series ......Page 616
13.1 Phase Plane, Paths, and Critical Points ......Page 640
13.2 Critical Points and Paths of Linear Systems ......Page 652
13.3 Critical Points and Paths of Nonlinear Systems ......Page 666
13.4 Limit Cycles and Periodic Solutions ......Page 700
13.5 The Method of Kryloff and Bogoliuboff ......Page 715
14.1 Some Basic Concepts and Examples ......Page 723
14.2 The Method of Separation of Variables ......Page 730
14.3 Canonical Forms of Second-Order Linear Equations with Constant Coefficients ......Page 751
14.4 An Initial-Value Problem; Characteristics ......Page 765
Appendix One - 2nd and 3rd Order Determinants ......Page 779
Appendix Two - exponential and logarithm tables ......Page 783
ANSWERS TO \r\nODD-NUMBERED EXERCISES ......Page 785
Suggested Reading ......Page 809
Index ......Page 811