Differential equations / Shepley L. Ross.

CONTENIDO
PART ONE. FUNDAMENTAL METHODS AND APPLICATIONS
One. Differential Equations and Their Solutions 3
1.1 Classification of Differential Equations; Their Origin and Application 3
1.2 Solutions 7
1.3 Initial- Value Problems, Boundary-Value Problems, and Existence of Solutions 15
Two. First-Order Equations for Which Exact Solutions Are Obtainable 25
2.1 Exact Differential Equations and Integrating Factors 25
2.2 Separable Equations and Equations Reducible to This Form 39
2.3 Linear Equations and Bernoulli Equations 49
2.4 Special Integrating Factors and Transformations 61
Three. Applications of First-Order Equations 70
3.1 Orthogonal and Oblique Trajectories 70
3.2 Problems in Mechanics 77
3.3 Rate Problems 89
Four. Explicit Methods of Solving Higher-Order Linear Differential Equations 102
4.1 Basic Theory of Linear Differential Equations 102
4.2 The Homogeneous Linear Equation with Constant Coefficients 125
4.3 The Method of Undetermined Coefficients 137
4.4 Variation of Parameters 155
4.5 The Cauchy-Euler Equation 164
4.6 Statements and Proofs of Theorems on the Second-Order Homogeneous Linear Equation 170
Five. Applications of Second-Order Linear Differential Equations with Constant Coefficients 179
5.1 The Differential Equation of the Vibrations of a Mass on a Spring 179
5.2 Free, Undamped Motion 182
5.3 Free, Damped Motion 189
5.4 Forced Motion 199
5.5 Resonance Phenomena 206
5.6 Electric Circuit Problems 211
Six. Series Solutions of Linear Differential Equations 221
6.1 Power Series Solutions About an Ordinary Point 221
6.2 Solutions About Singular Points; The Method of Frobenius 233
6.3 Bessel's Equation and Bessel Functions 252
Seven. Systems of Linear Differential Equations 264
7.1 Differential Operators and an Operator Method 264
7.2 Applications 278
7.3 Basic Theory of Linear Systems in Normal Form: Two Equations in Two Unknown Functions 290
7.4 Homogeneous Linear Systems with Constant Coefficients: Two Equations in Two Unknown Functions 301
7.5 Matrices and Vectors 312
7.6 The Matrix Method for Homogeneous Linear Systems with Constant Coefficients: Two Equations in Two Unknown Functions 346
7.7 The Matrix Method for Homogeneous Linear Systems with Constant Coefficients: n Equations in n Unknown Functions 355
Eight. Approximate Methods of Solving First-Order Equations 377
8.1 Graphical Methods 377
8.2 Power Series Methods 384
8.3 The Method of Successive Approximations 390
8.4 Numerical Methods 394
Nine. The Laplace Transform 411
9.1 Definition, Existence, and Basic Properties of the Laplace Transform 411
9.2 The Inverse Transform and the Convolution 431
9.3 Laplace Transform Solution of Linear Differential Equations with Constant Coefficients 441
9.4 Laplace Transform Solution of Linear Systems 453
PART TWO. FUNDAMENTAL THEORY AND FURTHER METHODS
Ten. Existence and Uniqueness Theory 461
10.1 Some Concepts from Real Function Theory 461
10.2 The Fundamental Existence and Uniqueness Theorem 473
10.3 Dependence of Solutions on Initial Conditions and on the Function 488
10.4 Existence and Uniqueness Theorems for Systems and Higher-Order Equations 495
Eleven. The Theory of Linear Differential Equations 505
11.1 Introduction 505
11.2 Basic Theory of the Homogeneous Linear System 510
11.3 Further Theory of the Homogeneous Linear System 522
11.4 The Nonhomogeneous Linear System 533
11.5 Basic Theory of the nth-Order Homogeneous Linear Differential Equation 543
11.6 Further Properties of the nth-Order Homogeneous Linear Differential Equation 558
11.7 The nth-Order Nonhomogcneous Linear Equation 569
11.8 Sturm Theory 573
Twelve. Sturm-Liouville Boundary-Value Problems and Fourier Series 588
12.1 Sturm-Liouville Problems 588
12.2 Orthogonality of Characteristic Functions 597
12.3 The Expansion of a Function in a Series of Orthonorma Functions 601
12.4 Trigonometric Fourier Series 608
Thirteen. Nonlinear Differential Equations 632
13.1 Phase Plane, Paths, and Critical Points 632
13.2 Critical Points and Paths of Linear Systems 644
13.3 Critical Points and Paths of Nonlinear Systems 658
13.4 Limit Cycles and Periodic Solutions 692
13.5 The Method of Kryloff and Bogoliuboff 707
Fourteen. Partial Differential Equations 715
14.1 Some Basic Concepts and Examples 715
14.2 The Method of Separation of Variables 722
14.3 Canonical Forms of Second-Order Linear Equations with Constant Coefficients 743
14.4 An Initial Value Problem; Characteristics 757
Appendices 771
Answers 777
Suggested Reading 801
Index 803