SOLUTIONS MANUAL for Differential Equations and Boundary Value Problems Computing and Modeling, 6th

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INSTRUCTOR’S SOLUTIONS MANUAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS COMPUTING AND MODELING SIXTH EDITION

C. Henry Edwards David E. Penney David Calvis

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Contents 1

First-Order Differential Equations

1.1 1.2 1.3 1.4 1.5 1.6

Differential Equations and Mathematical Models Integrals as General and Particular Solutions Slope Fields and Solution Curves Separable Equations and Applications Linear First-Order Equations Substitution Methods and Exact Equations Chapter 1 Review Problems

2

Mathematical Models and Numerical Methods

2.1 2.2 2.3 2.4 2.5 2.6

Population Models Equilibrium Solutions and Stability Acceleration-Velocity Models Numerical Approximation: Euler's Method A Closer Look at the Euler Method The Runge-Kutta Method

3

Linear Equations of Higher Order

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction: Second-Order Linear Equations General Solutions of Linear Equations Homogeneous Equations with Constant Coefficients Mechanical Vibrations Nonhomogeneous Equations and Undetermined Coefficients Forced Oscillations and Resonance Electrical Circuits Endpoint Problems and Eigenvalues

4

Introduction to Systems of Differential Equations

4.1 4.2 4.3

First-Order Systems and Applications The Method of Elimination Numerical Methods for Systems

5

Linear Systems of Differential Equations

5.1 5.2 5.3 5.4 5.5 5.6 5.7

Matrices and Linear Systems The Eigenvalue Method for Homogeneous Linear Systems Solution Curves of Linear Systems Second-Order Systems and Mechanical Applications Multiple Eigenvalue Solutions Matrix Exponentials and Linear Systems Nonhomogeneous Linear Systems

1 8 16 27 44 62 86

100 116 127 137 144 155

167 174 182 190 201 214 227 234

241 250 270

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280 288 313 319 331 345 355

6

Nonlinear Systems and Phenomena

6.1 6.2 6.3 6.4 6.5

Stability and the Phase Plane Linear and Almost Linear Systems Ecological Applications: Predators and Competitors Nonlinear Mechanical Systems Chaos in Dynamical Systems

7

Laplace Transform Methods

7.1 7.2 7.3 7.4 7.5 7.6

Laplace Transforms and Inverse Transforms Transformation of Initial Value Problems Translation and Partial Fractions Derivatives, Integrals, and Products of Transforms Periodic and Piecewise Continuous Input Functions Impulses and Delta Functions

8

Power Series Methods

8.1 8.2 8.3 8.4 8.5 8.6

Introduction and Review of Power Series Series Solutions Near Ordinary Points Regular Singular Points Method of Frobenius—The Exceptional Cases Bessel’s Equation Applications of Bessel Functions

9

Fourier Series Methods and Partial Differential Equations

9.1 9.2 9.3 9.4 9.5 9.6 9.7

Periodic Functions and Trigonometric Series General Fourier Series and Convergence Fourier Sine and Cosine Series Applications of Fourier Series Heat Conduction and Separation of Variables Vibrating Strings and the One-Dimensional Wave Equation Steady-State Temperature and Laplace’s Equation

10

Eigenvalue Methods and Boundary Value Problems

10.1 10.2 10.3 10.4 10.5

Sturm-Liouville Problems and Eigenfunction Expansions Applications of Eigenfunction Series Steady Periodic Solutions and Natural Frequencies Cylindrical Coordinate Problems Higher-Dimensional Phenomena

363 372 389 404 415

422 427 436 444 451 464

473 479 492 505 514 522

527 537 551 565 571 577 585

596 607 619 631 643

Appendix Existence and Uniqueness of Solutions

644

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Preface This is a solutions manual to accompany the textbook DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS: Computing and Modeling (6th edition, 2023) by C. Henry Edwards, David E. Penney, and David T. Calvis. We include solutions to most of the odd problems in the text. The corresponding Student Solutions Manual contains solutions to most of the odd-numbered solutions in the text.

Our goal is to support teaching of the subject of elementary differential equations in every way that we can.

We therefore invite comments and suggested improvements for future printings of this manual, as well as advice regarding features that might be added to increase its usefulness in subsequent editions. Additional supplementary material can be found at our textbook web site listed below.

Henry Edwards David Calvis h.edwards@mindspring.com dcalvis@bw.edu

http://www.pearsonhighered.com/irc/

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