Student Resource Manual to accompany

Work more effectively and gauge your progress along the way! This Student Resource Manual contains worked-out solutions to approximately half of the problems in Borrelli's Differential Equations, 2 nd Edition. In addition to problem solutions, it offers graphs, suggestions for students and additional resource material.With the modeling and graphical visualization as the central approach, Borrelli's Differential Equations, 2 nd Edition introduces differential systems and numerical methods early on and encourages the use of numerical solvers from the very start. It covers modern topics such as sensitivity, long-term behavior, bifurcation, and chaos together with the basic solution formula techniques and theory.

Robert L. Borrelli is the author of Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e, published by Wiley. Courtney S. Coleman is the author of Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e, published by Wiley.

1 Modeling and Differential Equations

1

(29)

1.1 The Modeling Approach

1

(4)

1.2 A Modeling Adventure

5

(3)

1.3 Models and Initial Value Problems

8

(7)

1.4 The Modeling Process: Differential Systems

15

(15)

Spotlight on Modeling: Radiocarbon Dating

19

(6)

Spotlight on Modeling: Cold Medication I

25

(5)

2 First Order Differential Equations

2.1 Linear Differential Equations

30

(3)

2.2 Linear Differential Equations: Qualitative Analysis

33

(5)

2.3 Existence and Uniqueness of Solutions

38

(5)

2.4 Visualizing Solution Curves: Slope Fields

43

(7)

2.5 Separable Differential Equations: Planar Systems

50

(5)

2.6 A Predator-Prey Model: the Lotka-Volterra System

55

(4)

2.7 Extension of Solutions: Long-Term Behavior

59

(5)

2.8 Qualitative Analysis: State Lines, Sign Analysis

64

(5)

2.9 Bifurcations: A Harvested Logistic Model

69

(29)

Spotlight on Approximate Numerical Solutions

73

(3)

Spotlight on Computer Implementation

76

(6)

Spotlight on Steady States: Linear ODEs

82

(4)

Spotlight on Modeling: Cold Medication II

86

(1)

Spotlight on Change of Variables: Pursuit Models

87

(9)

Spotlight on Continuity in the Data

96

(2)

3 Second Order Differential Equations

98

(52)

3.1 Models of Springs

99

(5)

3.2 Undriven Constant-Coefficient Linear Differential Equations

104

(4)

3.3 Visualizing Graphs of Solutions: Direction Fields

108

(5)

3.4 Periodic Solutions: Simple Harmonic Motion

113

(3)

3.5 Driven Linear ODEs: Undetermined Coefficients I

116

(3)

3.6 Driven Linear ODEs: Undetermined Coefficients II

119

(8)

3.7 Theory of Second-Order Linear Differential Equations

127

(7)

3.8 Nonlinear Second-Order Differential Equations

134

(16)

Spotlight on Modeling: Vertical Motion

142

(5)

Spotlight on Modeling: Shock Absorbers

147

(1)

Spotlight on Einstein's Field Equations

147

(3)

4 Applications of Second Order Differential Equations

150

(16)

4.1 Newton's Laws: The Pendulum

150

(5)

4.2 Beats and Resonance

155

(3)

4.3 Frequency Response Modeling

158

(4)

4.4 Electrical Circuits

162

(4)

Spotlight on Modeling: Tuning a Circuit

165

(1)

5 The Laplace Transform

166

(25)

5.1 The Laplace Transform: Solving IVPs

167

(4)

5.2 Working with the Transform

171

(4)

5.3 Transforms of Periodic Functions

175

(5)

5.4 Convolution

180

(11)

Spotlight on the Delta Function

184

(4)

Spotlight on Modeling: Time Delays and Collisions

188

(3)

6 Linear Systems of Differential Equations

191

(53)

6.1 Compartment Models: Tracking Lead

192

(7)

6.2 Eigenvalues, Eigenvectors, and Eigenspaces of Matrices

199

(3)

6.3 Undriven Linear Differential Systems: Real Eigenvalues

202

(6)

6.4 Undriven Linear Systems: Complex Eigenvalues

208

(7)

6.5 Orbital Portraits for Planar Systems

215

(5)

6.6 Driven Systems: The Matrix Exponential

220

(9)

6.7 Steady States

229

(4)

6.8 The Theory of General Linear Systems

233

(11)

Spotlight on Vectors, Matrices, Independence

236

(4)

Spotlight on Linear Algebraic Equations

240

(2)

Spotlight on Bifurcations: Sensitivity

242

(2)

7 Nonlinear Differential Systems

244

(18)

7.1 Chemical Kinetics: The Fundamental Theorem

245

(4)

7.2 Properties of Autonomous Systems, Direction Fields

249

(5)

7.3 Interacting Species: Cooperation, Competition

254

(8)

Spotlight on Modeling: Destructive Competition

260

(1)

Spotlight on Modeling: Bifurcation and Sensitivity

261

(1)

8 Stability

262

(23)

8.1 Stability and Linear Autonomous Systems

263

(4)

8.2 Stability and Nonlinear Autonomous Systems

267

(8)

8.3 Conservative Systems

275

(10)

Spotlight on Lyapunov Functions

279

(5)

Spotlight on Rotating Bodies

284

(1)

9 Nonlinear Systems: Cycles and Chaos

285

(28)

9.1 Cycles

286

(9)

9.2 Solution Behavior in Planar Autonomous Systems

295

(5)

9.3 Bifurcations

300

(7)

9.4 Chaos

307

(7)

Spotlight on Chaotic Systems

309

(4)

10 Fourier Series and Partial Differential Equations

313

(37)

10.1 Vibrations of a Guitar String

314

(7)

10.2 Fourier Trigonometric Series

321

(5)

10.3 Half-Range and Exponential Fourier Series

326

(3)

10.4 Temperature in a Thin Rod

329

(4)

10.5 Sturm-Liouville Problems

333

(4)

10.6 The Method of Eigenfunction Expansions

337

(14)

Spotlight on Decay Estimates

342

(1)

Spotlight on the Optimal Depth for a Wine Cellar

342

(1)

Spotlight on Approximation of Functions

343

(7)

11 Series Solutions

350

(21)

11.1 The Method of Power Series

351

(4)

11.2 Series Solutions near an Ordinary Point

355

(4)

11.3 Regular Singular Points: The Euler ODE

359

(2)

11.4 Series Solutions near Regular Singular Points

361

(10)

Spotlight on Legendre Polynomials

365

(3)

Spotlight on Bessel Functions

368

(3)

A. Basic Theory of Initial Value Problems

371

A.1 Uniqueness

371

(1)

A.2 The Picard Process for Solving an Initial Value Problem

372

(3)

A.3 Extension of Solutions

375

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Student Resource Manual to accompany Differential Equations: A Modeling Perspective, 2e

Summary

Author Biography

Table of Contents

Supplemental Materials