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Solutions for A Course in Ordinary Differential Equations, 2nd Edition by Wirkus (All Chapters included)
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  • A Course In Ordinary Differential Equations 2e

Complete Solutions Manual for A Course in Ordinary Differential Equations, 2nd Edition by Stephen A. Wirkus; Randall J. Swift ; ISBN13: 9781032917498.....(Full Chapters are included)...1.Traditional First-Order Differential Equations 2.Geometrical and Numerical Methods for First-Order Equations 3...

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  • November 14, 2024
  • 296
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solutions MANUAL FOR
A Course in
Ordinary
Differential
Equations, Second
Edition
by

Stephen A. Wirkus and
Randall J. Swift

** Immediate Download
** Swift Response
** All Chapters included




K14712_SM_Cover.indd 1 05/11/

, Contents


1 Traditional First-Order Differential Equations 1
1.1 Introduction to First-Order Differential Equations . . . . . . 1
1.2 Separable Differential Equations . . . . . . . . . . . . . . . . 6
1.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Some Physical Models Arising as Seperable Equations . . . . 22
1.5 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 Special Integrating Factors and Substitution Methods . . . . 35
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2 Geometrical and Numerical Methods for First-Order Equa-
tions 59
2.1 Direction Fields—the Geometry of Differential Equations . . 59
2.2 Existence and Uniqueness for First-Order Equations . . . . . 65
2.3 First-Order Autonomous Equations—Geometrical Insight . . 67
2.4 Modeling in Population Biology . . . . . . . . . . . . . . . . 83
2.5 Numerical Approximation: Euler and Runge-Kutta Methods 86
2.6 An Introduction to Autonomous Second-Order Equations . . 91
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3 Elements of Higher-Order Linear Equations 101
3.1 Introduction to Higher-Order Equations . . . . . . . . . . . . 101
3.2 Linear Independence and the Wronskian . . . . . . . . . . . 104
3.3 Reduction of Order—The Case of n = 2 . . . . . . . . . . . . 110
3.4 Numerical Considerations for nth-Order Equations . . . . . . 113
3.5 Essential Topics from Complex Variables . . . . . . . . . . . 119
3.6 Homogeneous Equations with Constant Coefficients . . . . . 123
3.7 Mechanical and Electrical Vibrations . . . . . . . . . . . . . 130
Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4 Techniques of Nonhomogeneous Higher-Order Linear Equa-
tions 141
4.1 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . 141
4.2 Method of Undetermined Coefficients via Superposition . . . 144
4.3 Method of Undetermined Coefficients via Annihilation . . . . 152
4.4 Exponential Response and Complex Replacement . . . . . . 161
4.5 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . 166
4.6 Cauchy-Euler (Equidimensional) Equation . . . . . . . . . . 177


v




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, vi

4.7 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 183
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5 Fundamentals of Systems of Differential Equations 189
5.1 Useful Terminology . . . . . . . . . . . . . . . . . . . . . . . 189
5.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . 192
5.3 Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . 193
5.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 197
5.5 A General Method, Part I: Solving Systems with Real & Dis-
tinct or Complex Eigenvalues . . . . . . . . . . . . . . . . . . 202
5.6 A General Method, Part II: Solving Systems with Repeated
Real Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.7 Matrix Exponentials . . . . . . . . . . . . . . . . . . . . . . . 210
5.8 Solving Linear Nonhomogeneous Systems of Equations . . . 212
Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6 Geometrical Approaches and Applications of Systems of Dif-
ferential Equations 223
6.1 An Introduction to the Phase Plane . . . . . . . . . . . . . . 223
6.2 Nonlinear Equations and Phase Plane Analysis . . . . . . . . 227
6.3 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.4 Epidemiological Models . . . . . . . . . . . . . . . . . . . . . 235
6.5 Models in Ecology . . . . . . . . . . . . . . . . . . . . . . . . 239
Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 244

7 Laplace Transforms 249
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.2 Fundamentals of the Laplace Transform . . . . . . . . . . . . 250
7.3 The Inverse Laplace Transform . . . . . . . . . . . . . . . . . 252
7.4 Translated Functions, Delta Function, and Periodic Functions 254
7.5 The s-Domain and Poles . . . . . . . . . . . . . . . . . . . . 256
7.6 Solving Linear Systems using Laplace Transforms . . . . . . 259
7.7 The Convolution . . . . . . . . . . . . . . . . . . . . . . . . . 259
Chapter 7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8 Series Methods 267
8.1 Power Series Representations of Functions . . . . . . . . . . 267
8.2 The Power Series Method . . . . . . . . . . . . . . . . . . . . 270
8.3 Ordinary and Singular Points . . . . . . . . . . . . . . . . . . 271
8.4 The Method of Frobenius . . . . . . . . . . . . . . . . . . . . 272
8.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . 274
Chapter 8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . 277

A An Introduction to MATLAB, Maple, and Mathematica 279




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, B Selected Topics from Linear Algebra 281
B.1 A Primer on Matrix Algebra . . . . . . . . . . . . . . . . . . 281
B.2 Matrix Inverses, and Cramer’s Rule . . . . . . . . . . . . . . 283
B.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . 287
B.4 Coordinates and Change of Basis . . . . . . . . . . . . . . . 289




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