Instructor’s Solutions Manual — Elementary Differential Equations with Boundary Value Problems (Classic Version), 6th Edition — C. Henry Edwards, David E. Penney, David T. Calvis

Differential Equations and Boundary Value
Problems: Computing and Modeling – 6th
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Edition
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INSTRUCTOR’S
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SOLUTIONS
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MANUAL
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C. Henry Edwards
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David E. Penney
David T. Calvis
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Comprehensive Solutions Manual for Instructors and
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Students
© C. Henry Edwards, David E. Penney & David T. Calvis
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All rights reserved. Reproduction or distribution without permission is prohibited.




©STUDYSTREAM

, Contents
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical Models 1
1.2 Integrals as General and Particular Solutions 8
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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
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Chapter 1 Review Problems 86

2 Mathematical Models and Numerical Methods
2.1 Population Models 100
2.2 Equilibrium Solutions and Stability 116
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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
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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
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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
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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
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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
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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
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, 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
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6.5 Chaos in Dynamical Systems 415

7 Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms 422
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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
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8 Power Series Methods
8.1 Introduction and Review of Power Series 473
8.2 Series Solutions Near Ordinary Points 479
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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
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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
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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
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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
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10.5 Higher-Dimensional Phenomena 643

Appendix
Existence and Uniqueness of Solutions 644
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, CHAPTER 1
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FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1
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DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS

The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of dif-
ferential equations, and to show the student what is meant by a solution of a differential equation.
Also, the use of differential equations in the mathematical modeling of real-world phenomena is
outlined.
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Problems 1-12 are routine verifications by direct substitution of the suggested solutions into the
given differential equations. We include here just some typical examples of such verifications.
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3. If y1  cos 2 x and y2  sin 2 x , then y1   2sin 2 x y2  2 cos 2 x , so
y1  4 cos 2 x  4 y1 and y2  4sin 2 x  4 y2 . Thus y1  4 y1  0 and y2  4 y2  0 .

4. If y1  e3 x and y2  e 3 x , then y1  3 e3 x and y2   3 e 3 x , so y1  9e3 x  9 y1 and
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y2  9e 3 x  9 y2 .

5. If y  e x  e  x , then y  e x  e  x , so y   y   e x  e  x    e x  e  x   2 e  x . Thus
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y  y  2 e  x .

6. If y1  e 2 x and y2  x e 2 x , then y1   2 e 2 x , y1  4 e 2 x , y2  e 2 x  2 x e 2 x , and
y2   4 e 2 x  4 x e 2 x . Hence
y1  4 y1  4 y1   4 e 2 x   4  2 e 2 x   4  e 2 x   0
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and
y2  4 y2  4 y2    4e 2 x
 4 x e 2 x   4  e 2 x  2 x e 2 x   4  x e 2 x   0.
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8. If y1  cos x  cos 2 x and y2  sin x  cos 2 x , then y1   sin x  2sin 2 x,
y1   cos x  4 cos 2 x, y2  cos x  2sin 2 x , and y2   sin x  4 cos 2 x. Hence
y1  y1    cos x  4 cos 2 x    cos x  cos 2 x   3cos 2 x
and
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y2  y2    sin x  4 cos 2 x    sin x  cos 2 x   3cos 2 x.



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