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Differential Equations and Linear Algebra, 4th Edition

Name: Differential Equations and Linear Algebra, 4th Edition
ISBN: 9780136739692

by C. Henry Edwards, David E. Penney, David T. Calvis, David Calvis

January 2017

Intermediate to advanced

768 pages

27h 45m

English

Pearson

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Differential Equations & Linear Algebra
Contents
Application Modules
Preface
Principal Features of This RevisionFeatures of This TextSupplements
1 First-Order Differential Equations
1.1 Differential Equations and Mathematical ModelsDifferential Equations and Mathematical ModelsMathematical ModelsExamples and TerminologySolutionSolution1.1 ProblemsDifferential Equations as Models
1.2 Integrals as General and Particular Solutions
SolutionVelocity and AccelerationSolutionPhysical UnitsVertical Motion with Gravitational AccelerationA Swimmer’s Problem1.2 ProblemsVelocity Given Graphically
1.3 Slope Fields and Solution Curves
Slope Fields and Graphical SolutionsSolutionApplications of Slope FieldsExistence and Uniqueness of Solutions1.3 Problems1.3 Application Computer-Generated Slope Fields and Solution Curves
1.4 Separable Equations and Applications
SolutionImplicit, General, and Singular SolutionsSolutionNatural Growth and DecayThe Natural Growth EquationSolutionSolutionCooling and HeatingSolutionTorricelli’s LawSolution1.4 ProblemsTorricelli’s Law1.4 Application The Logistic Equation
1.5 Linear First-Order Equations
SolutionSolutionA Closer Look at the MethodSolutionMixture ProblemsSolutionSolution1.5 ProblemsMixture ProblemsPolluted Reservoir1.5 Application Indoor Temperature Oscillations
1.6 Substitution Methods and Exact Equations
SolutionHomogeneous EquationsSolutionSolutionBernoulli EquationsFlight TrajectoriesSolutionExact Differential EquationsSolutionReducible Second-Order EquationsSolutionSolution1.6 Problems1.6 Application Computer Algebra Solutions

Chapter 1 Summary
Chapter 1 Review Problems
2 Mathematical Models and Numerical Methods
2.1 Population ModelsBounded Populations and the Logistic EquationLimiting Populations and Carrying CapacitySolutionHistorical NoteMore Applications of the Logistic EquationSolutionDoomsday versus ExtinctionSolution2.1 Problems2.1 Application Logistic Modeling of Population Data
2.2 Equilibrium Solutions and Stability
Stability of Critical PointsHarvesting a Logistic PopulationBifurcation and Dependence on Parameters2.2 ProblemsConstant-Rate Harvesting
2.3 Acceleration–Velocity Models
Resistance Proportional to VelocitySolutionResistance Proportional to Square of VelocityVariable Gravitational AccelerationSolutionEscape Velocity2.3 Problems2.3 Application Rocket PropulsionConstant ThrustNo ResistanceFree Space
2.4 Numerical Approximation: Euler’s Method
SolutionLocal and Cumulative ErrorsA Word of CautionSolution2.4 Problems2.4 Application Implementing Euler’s MethodFamous Numbers Investigation
2.5 A Closer Look at the Euler Method
Solution
An Improvement in Euler’s Method
Answer2.5 ProblemsDecreasing Step Size2.5 Application Improved Euler ImplementationFamous Numbers RevisitedLogistic Population InvestigationPeriodic Harvesting and Restocking
2.6 The Runge–Kutta Method
Solution2.6 ProblemsVelocity-Acceleration Problems2.6 Application Runge–Kutta ImplementationFamous Numbers Revisited, One Last TimeThe Skydiver’s Descent
3 Linear Systems and Matrices
3.1 Introduction to Linear SystemsTwo Equations in Two UnknownsThree PossibilitiesThe Method of EliminationThree Equations in Three UnknownsSolutionSolutionA Differential Equations Application3.1 Problems
3.2 Matrices and Gaussian Elimination
Coefficient MatricesElementary Row OperationsEchelon MatricesGaussian Elimination3.2 Problems3.2 Application Automated Row Reduction
3.3 Reduced Row-Echelon Matrices
SolutionThe Three PossibilitiesHomogeneous SystemsEqual Numbers of Equations and Variables3.3 Problems3.3 Application Automated Row Reduction
3.4 Matrix Operations
VectorsMatrix MultiplicationMatrix EquationsMatrix Algebra3.4 Problems
3.5 Inverses of Matrices
The Inverse Matrix A-1How to Find A-1SolutionMatrix EquationsSolutionNonsingular Matrices3.5 Problems3.5 Application Automated Solution of Linear Systems
3.6 Determinants
Higher-Order DeterminantsRow and Column PropertiesThe Transpose of a MatrixDeterminants and InvertibilityCramer’s Rule for n×n SystemsSolutionInverses and the Adjoint MatrixSolutionComputational Efficiency3.6 Problems
3.7 Linear Equations and Curve Fitting
SolutionModeling World Population GrowthGeometric ApplicationsSolutionSolution3.7 ProblemsPopulation Modeling
4 Vector Spaces
4.1 The Vector Space R3 The Vector Space R2 SolutionLinear Independence in R3Basis Vectors in R3Subspaces of R34.1 Problems
4.2 The Vector Space Rn and Subspaces
Definition of a Vector SpaceSubspaces4.2 Problems
4.3 Linear Combinations and Independence of Vectors
Linear Independence4.3 Problems
4.4 Bases and Dimension for Vector Spaces
Bases for Solution SpacesSolution4.4 Problems
4.5 Row and Column Spaces
Row Space and Row RankColumn Space and Column RankSolutionRank and DimensionNonhomogeneous Linear Systems4.5 Problems
4.6 Orthogonal Vectors in Rn
SolutionOrthogonal Complements4.6 Problems
4.7 General Vector Spaces
Function SpacesSolutionSolutionSolution Spaces of Differential Equations4.7 Problems
5 Higher-Order Linear Differential Equations
5.1 Introduction: Second-Order Linear EquationsA Typical ApplicationHomogeneous Second-Order Linear EquationsSolutionLinearly Independent SolutionsGeneral SolutionsLinear Second-Order Equations with Constant CoefficientsSolution5.1 Problems5.1 Application Plotting Second-Order Solution Families
5.2 General Solutions of Linear Equations
Existence and Uniqueness of SolutionsLinearly Independent SolutionsSolutionSolutionGeneral SolutionsNonhomogeneous EquationsSolution5.2 Problems5.2 Application Plotting Third-Order Solution Families
5.3 Homogeneous Equations with Constant Coefficients
The Characteristic EquationDistinct Real RootsSolutionPolynomial Differential OperatorsRepeated Real RootsSolutionComplex-Valued Functions and Euler’s FormulaComplex RootsSolutionSolutionRepeated Complex RootsSolutionSolution5.3 Problems5.3 Application Approximate Solutions of Linear Equations
5.4 Mechanical Vibrations
The Simple PendulumFree Undamped MotionSolutionFree Damped MotionSolution5.4 ProblemsSimple PendulumFree Damped MotionDifferential Equations and Determinism
5.5 Nonhomogeneous Equations and Undetermined Coefficients
SolutionSolutionSolutionSolutionThe General ApproachSolutionSolutionSolutionThe Case of DuplicationSolutionSolutionSolutionVariation of ParametersSolution5.5 Problems5.5 Application Automated Variation of Parameters
5.6 Forced Oscillations and Resonance
Undamped Forced OscillationsSolutionBeatsResonanceModeling Mechanical SystemsSolutionSolutionDamped Forced OscillationsSolution5.6 ProblemsAutomobile Vibrations5.6 Application Forced Vibrations
6 Eigenvalues and Eigenvectors
6.1 Introduction to EigenvaluesThe Characteristic EquationSolutionSolutionEigenspacesSolution6.1 Problems
6.2 Diagonalization of Matrices
Similarity and Diagonalization6.2 Problems
6.3 Applications Involving Powers of Matrices
SolutionTransition MatricesPredator-Prey ModelsThe Cayley-Hamilton Theorem6.3 ProblemsPredator-Prey
7 Linear Systems of Differential Equations
7.1 First-Order Systems and ApplicationsInitial ApplicationsFirst-Order SystemsSolutionSimple Two-Dimensional SystemsLinear Systems7.1 Problems7.1 Application Gravitation and Kepler’s Laws of Planetary Motion
7.2 Matrices and Linear Systems
First-Order Linear SystemsIndependence and General SolutionsInitial Value Problems and Elementary Row OperationsSolutionNonhomogeneous Solutions7.2 Problems
7.3 The Eigenvalue Method for Linear Systems
The Eigenvalue MethodDistinct Real EigenvaluesSolutionCompartmental AnalysisSolutionComplex EigenvaluesSolutionSolution7.3 ProblemsCascading Brine TanksInterconnected Brine TanksOpen Three-Tank SystemClosed Three-Tank System7.3 Application Automatic Calculation of Eigenvalues and Eigenvectors
7.4 A Gallery of Solution Curves of Linear Systems
Systems of Dimension n=2Real EigenvaluesSaddle PointsNodes: Sinks and SourcesZero Eigenvalues and Straight-Line SolutionsRepeated Eigenvalues; Proper and Improper NodesThe Special Case of a Repeated Zero EigenvalueComplex Conjugate Eigenvalues and EigenvectorsPure Imaginary Eigenvalues: Centers and Elliptical OrbitsSolutionComplex Eigenvalues: Spiral Sinks and SourcesSolutionSolutionA 3-Dimensional Example7.4 Problems7.4 Application Dynamic Phase Plane Graphics
7.5 Second-Order Systems and Mechanical Applications*
Solution of Second-Order SystemsForced Oscillations and ResonancePeriodic and Transient Solutions7.5 ProblemsThe Two-Axle Automobile7.5 Application Earthquake-Induced Vibrations of Multistory Buildings
7.6 Multiple Eigenvalue Solutions
SolutionDefective EigenvaluesThe Case of Multiplicity k=2 SolutionGeneralized EigenvectorsSolutionThe General CaseAn ApplicationThe Jordan Normal FormThe General Cayley-Hamilton Theorem7.6 Problems7.6 Application Defective Eigenvalues and Generalized Eigenvectors
7.7 Numerical Methods for Systems
Euler Methods for SystemsThe Runge–Kutta Method and Second-Order EquationsHigher-Order SystemsSolutionVariable Step Size MethodsEarth–Moon Satellite Orbits7.7 ProblemsBatted Baseball
7.7 Application Comets and Spacecraft
Your Spacecraft LandingKepler’s Law of Planetary (or Satellite) MotionHalley’s CometYour Own Comet
8 Matrix Exponential Methods
8.1 Matrix Exponentials and Linear SystemsFundamental Matrix SolutionsSolutionExponential MatricesMatrix Exponential SolutionsSolutionGeneral Matrix ExponentialsSolution8.1 Problems8.1 Application Automated Matrix Exponential Solutions
8.2 Nonhomogeneous Linear Systems
Undetermined CoefficientsSolutionVariation of ParametersSolution8.2 ProblemsTwo Brine Tanks8.2 Application Automated Variation of Parameters
8.3 Spectral Decomposition Methods
The Case of Distinct EigenvaluesSecond-Order Linear SystemsThe General Case8.3 Problems
9 Nonlinear Systems and Phenomena
9.1 Stability and the Phase PlaneSolutionPhase PortraitsCritical Point BehaviorStabilityAsymptotic Stability9.1 Problems9.1 Application Phase Plane Portraits and First-Order Equations
9.2 Linear and Almost Linear Systems
Linearization Near a Critical PointIsolated Critical Points of Linear SystemsAlmost Linear SystemsSolutionSolution9.2 ProblemsBifurcations9.2 Application Phase Plane Portraits of Almost Linear Systems
9.3 Ecological Models: Predators and Competitors
Competing SpeciesInteractions of Logistic Populations9.3 ProblemsPredator–Prey SystemCompetition SystemCompetition SystemLogistic Prey PopulationDoomsday vs. Extinction9.3 Application Your Own Wildlife Conservation Preserve
9.4 Nonlinear Mechanical Systems
The Position–Velocity Phase PlaneDamped Nonlinear VibrationsThe Nonlinear PendulumPeriod of Undamped OscillationDamped Pendulum Oscillations9.4 ProblemsCritical Points for Damped PendulumCritical Points for Mass-Spring SystemCritical Points for Physical SystemsPeriod of Oscillation9.4 Application The Rayleigh, van der Pol, and FitzHugh-Nagumo EquationsRayleigh’s EquationVan der Pol’s EquationThe FitzHugh-Nagumo Equations
10 Laplace Transform Methods
10.1 Laplace Transforms and Inverse TransformsLinearity of TransformsInverse TransformsPiecewise Continuous FunctionsSolutionGeneral Properties of Transforms10.1 Problems10.1 Application Computer Algebra Transforms and Inverse Transforms
10.2 Transformation of Initial Value Problems
Solution of Initial Value ProblemsSolutionSolutionLinear SystemsSolutionThe Transform PerspectiveAdditional Transform TechniquesSolutionSolutionSolutionExtension of Theorem 110.2 Problems10.2 Application Transforms of Initial Value Problems
10.3 Translation and Partial Fractions
SolutionSolutionSolutionSolutionResonance and Repeated Quadratic FactorsSolutionSolution10.3 Problems10.3 Application Damping and Resonance Investigations
10.4 Derivatives, Integrals, and Products of Transforms
Differentiation of TransformsSolutionSolutionIntegration of TransformsSolutionSolution* Proofs of Theorems10.4 Problems
10.5 Periodic and Piecewise Continuous Input Functions
SolutionSolutionSolutionTransforms of Periodic FunctionsSolution10.5 Problems10.5 Application Engineering Functions
11 Power Series Methods
11.1 Introduction and Review of Power SeriesPower Series OperationsThe Power Series MethodSolutionShift of Index of SummationSolutionSolutionSolution11.1 Problems
11.2 Power Series Solutions
SolutionSolutionTranslated Series SolutionsSolutionTypes of Recurrence RelationSolutionThe Legendre Equation11.2 Problems11.2 Application Automatic Computation of Series Coefficients
11.3 Frobenius Series Solutions
Types of Singular PointsThe Method of FrobeniusSolutionFrobenius Series SolutionsSolutionSolutionWhen r1-r2 Is an IntegerSolutionSummary11.3 Problems11.3 Application Automating the Frobenius Series Method
11.4 Bessel Functions
The Case r=p > 0 > 0The Case r = -p < 0 < 0 The Gamma FunctionBessel Functions of the First KindBessel Functions of the Second KindBessel Function IdentitiesApplications of Bessel FunctionsSolutionSolution11.4 Problems
References for Further Study
APPENDIX A Existence and Uniqueness of Solutions
A.1 Existence of Solutions
A.2 Linear Systems
A.3 Local Existence
A.4 Uniqueness of Solutions
A.5 Well-Posed Problems and Mathematical Models
Problems
APPENDIX B Theory of Determinants
Determinants and Elementary Row Operations
Determinants and Invertibility
Cramer’s Rule and Inverse Matrices
Inverses and the Adjoint Matrix
Answers to Selected Problems
Chapter 1
Section 1.1
Section 1.2
Section 1.3
Section 1.4
Section 1.5
Section 1.6
Chapter 1 Review Problems
Chapter 2
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Section 2.6
Chapter 3
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Section 3.5
Section 3.6
Section 3.7
Chapter 4
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Section 4.6
Section 4.7
Chapter 5
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 5.5
Section 5.6
Chapter 6
Section 6.1
Section 6.2
Section 6.3
Chapter 7
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Section 7.5
Section 7.6
Section 7.7
Chapter 8
Section 8.1
Section 8.2
Section 8.3
Chapter 9
Section 9.1
Section 9.2
Section 9.3
Section 9.4
Chapter 10
Section 10.1
Section 10.2
Section 10.3
Section 10.4
Section 10.5
Chapter 11
Section 11.1
Section 11.2
Section 11.3
Section 11.4
Appendix A
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z

Content preview from Differential Equations and Linear Algebra, 4th Edition

7.3 The Eigenvalue Method for Linear Systems

We now introduce a powerful method for constructing the general solution of a homogeneous first-order system with constant coefficients,

\begin{array}{rcl} x_{1}^{′} & = & a_{11} x_{1} & + & a_{12} x_{2} & + & \dots & + & a_{1 n} x_{n}, \\ x_{2}^{′} & = & a_{21} x_{1} & + & a_{22} x_{2} & + & \dots & + & a_{2 n} x_{n}, \\ ⋮ \\ x_{n}^{′} & = & a_{n 1} x_{1} & + & a_{n 2} x_{2} & + & \dots & + & a_{n n} x_{n .} \end{array}

(1)

By Theorem 3 of Section 7.2, we know that it suffices to find n linearly independent solution vectors $x_{1}, x_{2}, \dots, x_{n}$ ; the linear combination

x (t) = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n}

(2)

with arbitrary coefficients will then be a general solution of the system in (1).

To search for the n needed linearly independent solution vectors, we proceed by analogy with the characteristic root method for solving a single ...

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Differential Equations and Linear Algebra, 4th Edition

by C. Henry Edwards, David E. Penney, David T. Calvis, David Calvis

7.3 The Eigenvalue Method for Linear Systems

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