Elementary Differential Equations and Boundary Value Problem

by Boyce, William E.; Diprima, Richard C.

• ISBN13: 9780471089551
• ISBN10: 0471089559
• Edition: 6th
• Format: Hardcover
• Copyright: 1996-08-01
• Publisher: John Wiley & Sons Inc

Summary

This book covers all the essential topics on differential equations, including series solutions, Laplace transforms, systems of equations, numerical methods and phase plane methods. Clear explanations are detailed with many current examples.

Table of Contents

Preface	vii
CHAPTER 1 Introduction	1	(14)
1.1 Classification of Differential Equations	1	(10)
1.2 Historical Remarks	11	(4)
CHAPTER 2 First Order Differential Equations	15	(106)
2.1 Linear Equations	15	(10)
2.2 Further Discussion of Linear Equations	25	(8)
2.3 Separable Equations	33	(7)
2.4 Differences Between Linear and Nonlinear Equations	40	(6)
2.5 Modeling with Linear Equations	46	(12)
2.6 Population Dynamics and Some Related Problems	58	(16)
2.7 Some Problems in Mechanics	74	(9)
2.8 Exact Equations and Integrating Factors	83	(7)
2.9 Homogeneous Equations	90	(4)
2.10 Miscellaneous Problems and Applications	94	(4)
2.11 The Existence and Uniqueness Theorem	98	(10)
2.12 First Order Difference Equations	108	(13)
CHAPTER 3 Second Order Linear Equations	121	(82)
3.1 Homogeneous Equations with Constant Coefficients	121	(9)
3.2 Fundamental Solutions of Linear Homogeneous Equations	130	(9)
3.3 Linear Independence and the Wronskian	139	(6)
3.4 Complex Roots of the Characteristic Equation	145	(8)
3.5 Repeated Roots; Reduction of Order	153	(9)
3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients	162	(10)
3.7 Variation of Parameters	172	(7)
3.8 Mechanical and Electrical Vibrations	179	(14)
3.9 Forced Vibrations	193	(10)
CHAPTER 4 Higher Order Linear Equations	203	(22)
4.1 General Theory of nth Order Linear Equations	203	(5)
4.2 Homogeneous Equations with Constant Coefficients	208	(8)
4.3 The Method of Undetermined Coefficients	216	(5)
4.4 The Method of Variation of Parameters	221	(4)
CHAPTER 5 Series Solutions of Second Order Linear Equations	225	(64)
5.1 Review of Power Series	225	(7)
5.2 Series Solutions near an Ordinary Point, Part I	232	(11)
5.3 Series Solutions near an Ordinary Point, Part II	243	(7)
5.4 Regular Singular Points	250	(5)
5.5 Euler Equations	255	(7)
5.6 Series Solutions near a Regular Singular Point, Part I	262	(5)
5.7 Series Solutions near a Regular Singular Point, Part II	267	(8)
5.8 Bessel's Equation	275	(14)
CHAPTER 6 The Laplace Transform	289	(46)
6.1 Definition of the Laplace Transform	289	(6)
6.2 Solution of Initial Value Problems	295	(11)
6.3 Step Functions	306	(7)
6.4 Differential Equations with Discontinuous Forcing Functions	313	(6)
6.5 Impulse Functions	319	(6)
6.6 The Convolution Integral	325	(10)
CHAPTER 7 Systems of First Order Linear Equations	335	(80)
7.1 Introduction	335	(9)
7.2 Review of Matrices	344	(9)
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors	353	(12)
7.4 Basic Theory of Systems of First Order Linear Equations	365	(5)
7.5 Homogeneous Linear Systems with Constant Coefficients	370	(11)
7.6 Complex Eigenvalues	381	(9)
7.7 Repeated Eigenvalues	390	(8)
7.8 Fundamental Matrices	398	(7)
7.9 Nonhomogeneous Linear Systems	405	(10)
CHAPTER 8 Numerical Methods	415	(44)
8.1 The Euler or Tangent Line Method	415	(8)
8.2 Errors in Numerical Procedures	423	(6)
8.3 Improvements on the Euler Method	429	(7)
8.4 The Runge-Kutta Method	436	(3)
8.5 Multistep Methods	439	(6)
8.6 More on Errors; Stability	445	(9)
8.7 Systems of First Order Equations	454	(5)
CHAPTER 9 Nonlinear Differential Equations and Stability	459	(84)
9.1 The Phase Plane; Linear Systems	459	(12)
9.2 Autonomous Systems and Stability	471	(9)
9.3 Almost Linear Systems	480	(12)
9.4 Competing Species	492	(12)
9.5 Predator-Prey Equations	504	(8)
9.6 Liapunov's Second Method	512	(10)
9.7 Periodic Solutions and Limit Cycles	522	(11)
9.8 Chaos and Strange Attractors; the Lorenz Equations	533	(10)
CHAPTER 10 Partial Differential Equations and Fourier Series	543	(78)
10.1 Separation of Variables; Heat Conduction	543	(9)
10.2 Fourier Series	552	(11)
10.3 The Fourier Convergence Theorem	563	(6)
10.4 Even and Odd Functions	569	(9)
10.5 Other Heat Conduction Problems	578	(12)
10.6 The Wave Equation; Vibrations of an Elastic String	590	(12)
10.7 Laplace's Equation	602	(10)
Appendix A. Derivation of the Heat Conduction Equation	612	(4)
Appendix B. Derivation of the Wave Equation	616	(5)
CHAPTER 11 Boundary Value Problems and Sturm-Liouville Theory	621	(62)
11.1 The Occurrence of Two Point Boundary Value Problems	621	(4)
11.2 Linear Homogeneous Boundary Value Problems; Eigenvalues and Eigenfunctions	625	(8)
11.3 Sturm-Liouville Boundary Value Problems	633	(13)
11.4 Nonhomogeneous Boundary Value Problems	646	(15)
11.5 Singular Sturm-Liouville Problems	661	(8)
11.6 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion	669	(6)
11.7 Series of Orthogonal Functions: Mean Convergence	675	(8)
Answers to Problems	683	(60)
Index	743