,Contents
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical Models 1
1.2 Integrals as General and Particular Solutions 8
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 27
1.5 Linear First-Order Equations 44
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 Mathematical Models and Numerical Methods
2.1 Population Models 100
2.2 Equilibrium Solutions and Stability 116
2.3 Acceleration-Velocity Models 127
2.4 Numerical Approximation: Euler's Method 137
2.5 A Closer Look at the Euler Method 144
2.6 The Runge-Kutta Method 155
3 Linear Equations of Higher Order
3.1 Introduction: Second-Order Linear Equations 167
3.2 General Solutions of Linear Equations 174
3.3 Homogeneous Equations with Constant Coefficients 182
3.4 Mechanical Vibrations 190
3.5 Nonhomogeneous Equations and Undetermined Coefficients 201
3.6 Forced Oscillations and Resonance 214
3.7 Electrical Circuits 227
3.8 Endpoint Problems and Eigenvalues 234
4 Introduction to Systems of Differential Equations
4.1 First-Order Systems and Applications 241
4.2 The Method of Elimination 250
4.3 Numerical Methods for Systems 270
5 Linear Systems of Differential Equations
5.1 Matrices and Linear Systems 280
5.2 The Eigenvalue Method for Homogeneous Linear Systems 288
5.3 Solution Curves of Linear Systems 313
5.4 Second-Order Systems and Mechanical Applications 319
5.5 Multiple Eigenvalue Solutions 331
5.6 Matrix Exponentials and Linear Systems 345
5.7 Nonhomogeneous Linear Systems 355
iii
Copyright © 2023 Pearson Education, Inc.
,6 Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane 363
6.2 Linear and Almost Linear Systems 372
6.3 Ecological Applications: Predators and Competitors 389
6.4 Nonlinear Mechanical Systems 404
6.5 Chaos in Dynamical Systems 415
7 Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms 422
7.2 Transformation of Initial Value Problems 427
7.3 Translation and Partial Fractions 436
7.4 Derivatives, Integrals, and Products of Transforms 444
7.5 Periodic and Piecewise Continuous Input Functions 451
7.6 Impulses and Delta Functions 464
8 Power Series Methods
8.1 Introduction and Review of Power Series 473
8.2 Series Solutions Near Ordinary Points 479
8.3 Regular Singular Points 492
8.4 Method of Frobenius—The Exceptional Cases 505
8.5 Bessel’s Equation 514
8.6 Applications of Bessel Functions 522
9 Fourier Series Methods and Partial Differential Equations
9.1 Periodic Functions and Trigonometric Series 527
9.2 General Fourier Series and Convergence 537
9.3 Fourier Sine and Cosine Series 551
9.4 Applications of Fourier Series 565
9.5 Heat Conduction and Separation of Variables 571
9.6 Vibrating Strings and the One-Dimensional Wave Equation 577
9.7 Steady-State Temperature and Laplace’s Equation 585
10 Eigenvalue Methods and Boundary Value Problems
10.1 Sturm-Liouville Problems and Eigenfunction Expansions 596
10.2 Applications of Eigenfunction Series 607
10.3 Steady Periodic Solutions and Natural Frequencies 619
10.4 Cylindrical Coordinate Problems 631
10.5 Higher-Dimensional Phenomena 643
Appendix
Existence and Uniqueness of Solutions 644
iv
Copyright © 2023 Pearson Education, Inc.
, 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
http://www.pearsonhighered.com/irc/
v
Copyright © 2023 Pearson Education, Inc.
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical Models 1
1.2 Integrals as General and Particular Solutions 8
1.3 Slope Fields and Solution Curves 16
1.4 Separable Equations and Applications 27
1.5 Linear First-Order Equations 44
1.6 Substitution Methods and Exact Equations 62
Chapter 1 Review Problems 86
2 Mathematical Models and Numerical Methods
2.1 Population Models 100
2.2 Equilibrium Solutions and Stability 116
2.3 Acceleration-Velocity Models 127
2.4 Numerical Approximation: Euler's Method 137
2.5 A Closer Look at the Euler Method 144
2.6 The Runge-Kutta Method 155
3 Linear Equations of Higher Order
3.1 Introduction: Second-Order Linear Equations 167
3.2 General Solutions of Linear Equations 174
3.3 Homogeneous Equations with Constant Coefficients 182
3.4 Mechanical Vibrations 190
3.5 Nonhomogeneous Equations and Undetermined Coefficients 201
3.6 Forced Oscillations and Resonance 214
3.7 Electrical Circuits 227
3.8 Endpoint Problems and Eigenvalues 234
4 Introduction to Systems of Differential Equations
4.1 First-Order Systems and Applications 241
4.2 The Method of Elimination 250
4.3 Numerical Methods for Systems 270
5 Linear Systems of Differential Equations
5.1 Matrices and Linear Systems 280
5.2 The Eigenvalue Method for Homogeneous Linear Systems 288
5.3 Solution Curves of Linear Systems 313
5.4 Second-Order Systems and Mechanical Applications 319
5.5 Multiple Eigenvalue Solutions 331
5.6 Matrix Exponentials and Linear Systems 345
5.7 Nonhomogeneous Linear Systems 355
iii
Copyright © 2023 Pearson Education, Inc.
,6 Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane 363
6.2 Linear and Almost Linear Systems 372
6.3 Ecological Applications: Predators and Competitors 389
6.4 Nonlinear Mechanical Systems 404
6.5 Chaos in Dynamical Systems 415
7 Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms 422
7.2 Transformation of Initial Value Problems 427
7.3 Translation and Partial Fractions 436
7.4 Derivatives, Integrals, and Products of Transforms 444
7.5 Periodic and Piecewise Continuous Input Functions 451
7.6 Impulses and Delta Functions 464
8 Power Series Methods
8.1 Introduction and Review of Power Series 473
8.2 Series Solutions Near Ordinary Points 479
8.3 Regular Singular Points 492
8.4 Method of Frobenius—The Exceptional Cases 505
8.5 Bessel’s Equation 514
8.6 Applications of Bessel Functions 522
9 Fourier Series Methods and Partial Differential Equations
9.1 Periodic Functions and Trigonometric Series 527
9.2 General Fourier Series and Convergence 537
9.3 Fourier Sine and Cosine Series 551
9.4 Applications of Fourier Series 565
9.5 Heat Conduction and Separation of Variables 571
9.6 Vibrating Strings and the One-Dimensional Wave Equation 577
9.7 Steady-State Temperature and Laplace’s Equation 585
10 Eigenvalue Methods and Boundary Value Problems
10.1 Sturm-Liouville Problems and Eigenfunction Expansions 596
10.2 Applications of Eigenfunction Series 607
10.3 Steady Periodic Solutions and Natural Frequencies 619
10.4 Cylindrical Coordinate Problems 631
10.5 Higher-Dimensional Phenomena 643
Appendix
Existence and Uniqueness of Solutions 644
iv
Copyright © 2023 Pearson Education, Inc.
, 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
http://www.pearsonhighered.com/irc/
v
Copyright © 2023 Pearson Education, Inc.
