Elementary Differential Equations and Boundary Value Problems

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

• ISBN13: 9780471509981
• ISBN10: 0471509981
• Edition: 5th
• Format: Hardcover
• Copyright: 1992-01-01
• Publisher: John Wiley & Sons Inc

Summary

Details the methods for solving ordinary and partial differential equations. New material on limit cycles, the Lorenz equations and chaos has been added along with nearly 300 new problems. Also features expanded discussions of competing species and predator-prey problems plus extended treatment of phase plane analysis, qualitative methods and stability.

Author Biography

William E. Boyce is currently the Edward P. Hamilton Distinguished Professor Emeritus of Science Education (Department of Mathematical Sciences) at Rensselaer Richard C. DiPrima (deceased) received his B.S., M.S., and Ph.D. degrees in Mathematics from Carnegie-Mellon University. He joined the faculty of Rensselaer Polytechnic Institute after holding research positions at MIT, Harvard, and Hughes Aircraft. He held the Eliza Ricketts Foundation Professorship of Mathematics at Rensselaer

Table of Contents

Preface	p. vii
Introduction	p. 1
Some Basic Mathematical Models; Direction Fields	p. 1
Solutions of Some Differential Equations	p. 9
Classification of Differential Equations	p. 17
Historical Remarks	p. 23
First Order Differential Equations	p. 29
Linear Equations with Variable Coefficients	p. 29
Separable Equations	p. 40
Modeling with First Order Equations	p. 47
Differences Between Linear and Nonlinear Equations	p. 64
Autonomous Equations and Population Dynamics	p. 74
Exact Equations and Integrating Factors	p. 89
Numerical Approximations: Euler's Method	p. 96
The Existence and Uniqueness Theorem	p. 105
First Order Difference Equations	p. 115
Second Order Linear Equations	p. 129
Homogeneous Equations with Constant Coefficients	p. 129
Fundamental Solutions of Linear Homogeneous Equations	p. 137
Linear Independence and the Wronskian	p. 147
Complex Roots of the Characteristic Equation	p. 153
Repeated Roots; Reduction of Order	p. 160
Nonhomogeneous Equations; Method of Undetermined Coefficients	p. 169
Variation of Parameters	p. 179
Mechanical and Electrical Vibrations	p. 186
Forced Vibrations	p. 200
Higher Order Linear Equations	p. 209
General Theory of nth Order Linear Equations	p. 209
Homogeneous Equations with Constant Coeffients	p. 214
The Method of Undetermined Coefficients	p. 222
The Method of Variation of Parameters	p. 226
Series Solutions of Second Order Linear Equations	p. 231
Review of Power Series	p. 231
Series Solutions near an Ordinary Point, Part I	p. 238
Series Solutions near an Ordinary Point, Part II	p. 249
Regular Singular Points	p. 255
Euler Equations	p. 260
Series Solutions near a Regular Singular Point, Part I	p. 267
Series Solutions near a Regular Singular Point, Part II	p. 272
Bessel's Equation	p. 280
The Laplace Transform	p. 293
Definition of the Laplace Transform	p. 293
Solution of Initial Value Problems	p. 299
Step Functions	p. 310
Differential Equations with Discontinuous Forcing Functions	p. 317
Impulse Functions	p. 324
The Convolution Integral	p. 330
Systems of First Order Linear Equations	p. 339
Introduction	p. 339
Review of Matrices	p. 348
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors	p. 357
Basic Theory of Systems of First Order Linear Equations	p. 368
Homogeneous Linear Systems with Constant Coefficients	p. 373
Complex Eigenvalues	p. 384
Fundamental Matrices	p. 393
Repeated Eigenvalues	p. 401
Nonhomogeneous Linear Systems	p. 411
Numerical Methods	p. 419
The Euler or Tangent Line Method	p. 419
Improvements on the Euler Method	p. 430
The Runge-Kutta Method	p. 435
Multistep Methods	p. 439
More on Errors; Stability	p. 445
Systems of First Order Equations	p. 455
Nonlinear Differential Equations and Stability	p. 459
The Phase Plane; Linear Systems	p. 459
Autonomous Systems and Stability	p. 471
Almost Linear Systems	p. 479
Competing Species	p. 491
Predator-Prey Equations	p. 503
Liapunov's Second Method	p. 511
Periodic Solutions and Limit Cycles	p. 521
Chaos and Strange Attractors; the Lorenz Equations	p. 532
Partial Differential Equations and Fourier Series	p. 541
Two-Point Boundary Valve Problems	p. 541
Fourier Series	p. 547
The Fourier Convergence Theorem	p. 558
Even and Odd Functions	p. 564
Separation of Variables; Heat Conduction in a Rod	p. 573
Other Heat Conduction Problems	p. 581
The Wave Equation; Vibrations of an Elastic String	p. 591
Laplace's Equation	p. 604
Derivation of the Heat Conduction Equation	p. 614
Derivation of the Wave Equation	p. 617
Boundary Value Problems and Sturm-Liouville Theory	p. 621
The Occurrence of Two Point Boundary Value Problems	p. 621
Sturm-Liouville Boundary Value Problems	p. 629
Nonhomogeneous Boundary Value Problems	p. 641
Singular Sturm-Liouville Problems	p. 656
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion	p. 663
Series of Orthogonal Functions: Mean Convergence	p. 669
Answers to Problems	p. 679
Index	p. 737
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