Greenberg, Michael D.
Ordinary Differential Equations. - 1 online resource (549 pages) - New York Academy of Sciences Ser. . - New York Academy of Sciences Ser. .
Cover -- Title Page -- Copyright -- Contents -- Preface -- 1 First-Order Differential Equations -- 1.1 Motivation And Overview -- 1.1.1 Introduction -- 1.1.2 Modeling -- 1.1.3 The order of a differential equation -- 1.1.4 Linear and nonlinear equations -- 1.1.5 Ourplan -- 1.1.6 Direction field -- 1.1.7 Computer software -- 1.2 Linear First-Order Equations -- 1.2.1 The simplest case -- 1.2.2 The homogeneous equation -- 1.2.3 Solving the full equation by the integrating factor method -- 1.2.4 Existence and uniqueness for the linear equation -- 1.3 Applications Of Linear First-Order Equations -- 1.3.1 Population dynamics -- exponential model -- 1.3.2 Radioactive decay -- carbon dating -- 1.3.3 Mixing problems -- a one-compartment model -- 1.3.4 The phase line, equilibrium points, and stability -- 1.3.5 Electrical circuits -- 1.4 Nonlinear First-Order Equations That Are Separable -- 1.5 Existence And Uniqueness -- 1.5.1 An existence and uniqueness theorem -- 1.5.2 Illustrating the theorem -- 1.5.3 Application to free fall -- physical significance of nonuniqueness -- 1.6 Applications Of Nonlinear First-Order Equations -- 1.6.1 The logistic model of population dynamics -- 1.6.2 Stability of equilibrium points and linearized stability analysis -- 1.7 Exact Equations And Equations That Can Be Made Exact -- 1.7.1 Exact differential equations -- 1.7.2 Making an equation exact -- integrating factors -- 1.8 Solution By Substitution -- 1.8.1 Bernoulli's equation -- 1.8.2 Homogeneous equations -- 1.9 Numerical Solution By Euler's Method -- 1.9.1 Euler's method -- 1.9.2 Convergence of Euler's method -- 1.9.3 Higher-order methods -- Chapter 1 Review -- 2 Higher-Order Linear Equations -- 2.1 Linear Differential Equations Of Second Order -- 2.1.1 Introduction -- 2.1.2 Operator notation and linear differential operators -- 2.1.3 Superposition principle. 2.2 Constant-Coefficient Equations -- 2.2.1 Constant coefficients -- 2.2.2 Seeking a general solution -- 2.2.3 Initial value problem -- 2.3 Complex Roots -- 2.3.1 Complex exponential function -- 2.3.2 Complex characteristic roots -- 2.4 Linear Independence -- Existence, Uniqueness, General Solution -- 2.4.1 Linear dependence and linear independence -- 2.4.2 Existence, uniqueness, and general solution -- 2.4.3 Abel's formula and Wronskian test for linear independence -- 2.4.4 Building a solution method on these results -- 2.5 Reduction Of Order -- 2.5.1 Deriving the formula -- 2.5.2 The method rather than the formula -- 2.5.3 About the method of reduction of order -- 2.6 Cauchy-Euler Equations -- 2.6.1 General solution -- 2.6.2 Repeated roots and reduction of order -- 2.6.3 Complex roots -- 2.7 The General Theory For Higher-Order Equations -- 2.7.1 Theorems for nth-order linear equations -- 2.7.2 Constant-coefficient equations -- 2.7.3 Cauchy-Euler equations -- 2.8 Nonhomogeneous Equations -- 2.8.1 General solution -- 2.8.2 The scaling and superposition of forcing functions -- 2.9 Particular Solution By Undetermined Coefficients -- 2.9.1 Undetermined coefficients -- 2.9.2 A special case -- the complex exponential method -- 2.10 Particular Solution By Variation Of Parameters -- 2.10.1 First-order equations -- 2.10.2 Second-order equations -- Chapter 2 Review -- 3 Applications Of Higher-Order Equations -- 3.1 Introduction -- 3.2 Linear Harmonic Oscillator -- Free Oscillation -- 3.2.1 Mass-spring oscillator -- 3.2.2 Undamped free oscillation -- 3.2.3 Pendulum -- 3.3 Free Oscillation With Damping -- 3.3.1 Underdamped -- 3.3.2 Critically damped -- 3.3.3 Overdamped -- 3.4 Forced Oscillation -- 3.4.1 Undamped, c = 0 -- 3.4.2 Damped, c > -- 0 -- 3.5 Steady-State Diffusion -- A Boundary Value Problem -- 3.5.1 Boundary value problems. existence and uniqueness -- 3.5.2 Steady-state heat conduction in a rod -- 3.6 Introduction To The Eigenvalue Problem -- Column Buckling -- 3.6.1 An eigenvalue problem -- 3.6.2 Application to column buckling -- Chapter 3 Review -- 4 Systems Of Linear Differential Equations -- 4.1 Introduction, And Solution By Elimination -- 4.1.1 Introduction -- 4.1.2 Physical examples -- 4.1.3 Solutions, existence, and uniqueness -- 4.1.4 Solution by elimination -- 4.1.5 Auxiliary variables -- 4.2 Application To Coupled Oscillators -- 4.2.1 Coupled oscillators -- 4.2.2 Reduction to first-order system by auxiliary variables -- 4.2.3 The free vibration -- 4.2.4 The forced vibration -- 4.3 N-Space And Matrices -- 4.3.1 Passage from 2-space to n-space -- 4.3.2 Matrix operators on vectors in n-space -- 4.3.3 Identity matrix and zero matrix -- 4.3.4 Relevance to systems of linear algebraic equations -- 4.3.5 Vector and matrix functions -- 4.4 Linear Dependence And Independence Of Vectors -- 4.4.1 Linear dependence of a set of constant vectors in n-space -- 4.4.2 Linear dependence of vector functions in n-space -- 4.5 Existence, Uniqueness, And General Solution -- 4.5.1 The key theorems -- 4.5.2 Illustrating the theorems -- 4.6 Matrix Eigenvalue Problem -- 4.6.1 The eigenvalue problem -- 4.6.2 Solving an eigenvalue problem -- 4.6.3 Complex eigenvalues and eigenvectors -- 4.7 Homogeneous Systems With Constant Coefficients -- 4.7.1 Solution by the method of assumed exponential form -- 4.7.2 Application to the two-mass oscillator -- 4.7.3 The case of repeated eigenvalues -- 4.7.4 Modifying the method if there are defective eigenvalues -- 4.7.5 Complex eigenvalues -- 4.8 Dot Product And Additional Matrix Algebra -- 4.8.1 More about n-space: dot product, norm, and angle -- 4.8.2 Algebra of matrix operators -- 4.8.3 Inverse matrix. 4.9 Explicit Solution Of x' = Ax And The Matrix Exponential Function -- 4.9.1 Matrix exponential solution -- 4.9.2 Getting the exponential matrix series into closed form -- 4.10 Nonhomogeneous Systems -- 4.10.1 Solution by variation of parameters -- 4.10.2 Constant coefficient matrix -- 4.10.3 Particular solution by undetermined coefficients -- Chapter 4 Review -- 5 Laplace Transform -- 5.1 Introduction -- 5.2 The Transform And Its Inverse -- 5.2.1 Laplace transform -- 5.2.2 Linearity property of the transform -- 5.2.3 Exponential order, piecewise continuity, and conditions for existence of the transform -- 5.2.4 Inverse transform -- 5.2.5 Introduction to the determination of inverse transforms -- 5.3 Application To The Solution Of Differential Equations -- 5.3.1 First-order equations -- 5.3.2 Higher-order equations -- 5.3.3 Systems -- 5.3.4 Application to a nonconstant-coefficient equation -- Bessel's equation -- 5.4 Discontinuous Forcing Functions -- Heaviside Step Function -- 5.4.1 Motivation -- 5.4.2 Heaviside step function and piecewise-defined functions -- 5.4.3 Transforms of Heaviside and time-delayed functions -- 5.4.4 Differential equations with piecewise-defined forcing functions -- 5.4.5 Periodic forcing functions -- 5.5 Convolution -- 5.5.1 Definition of Laplace convolution -- 5.5.2 Convolution theorem -- 5.5.3 Applications -- 5.5.4 Integro-differential equations and integral equations -- 5.6 Impulsive Forcing Functions -- Dirac Delta Function -- 5.6.1 Impulsive forces -- 5.6.2 Dirac delta function -- 5.6.3 The jump caused by the delta function -- 5.6.4 Caution -- 5.6.5 Impulse response function -- Chapter 5 Review -- 6 Series Solutions -- 6.1 Introduction -- 6.2 Power Series And Taylor Series -- 6.2.1 Power series -- 6.2.2 Manipulation of power series -- 6.2.3 Taylor series -- 6.3 Power Series Solution About A Regular Point. 6.3.1 Power series solution theorem -- 6.3.2 Applications -- 6.4 Legendre And Bessel Equations -- 6.4.1 Introduction -- 6.4.2 Legendre's equation -- 6.4.3 Bessel's equation -- 6.5 The Method of Frobenius -- 6.5.1 Motivation -- 6.5.2 Regular and irregular singular points -- 6.5.3 The method of Frobenius -- Chapter 6 Review -- 7 Systems Of Nonlinear Differential Equations -- 7.1 Introduction -- 7.2 The Phase Plane -- 7.2.1 Phase plane method -- 7.2.2 Application to nonlinear pendulum -- 7.2.3 Singular points and their stability -- 7.3 Linear Systems -- 7.3.1 Introduction -- 7.3.2 Purely imaginary eigenvalues (Center) -- 7.3.3 Complex conjugate eigenvalues (Spiral) -- 7.3.4 Real eigenvalues of the same sign (Node) -- 7.3.5 Real eigenvalues of opposite sign (Saddle) -- 7.4 Nonlinear Systems -- 7.4.1 Local linearization -- 7.4.2 Predator-prey population dynamics -- 7.4.3 Competing species -- 7.5 Limit Cycles -- 7.6 Numerical Solution Of Systems By Euler's Method -- 7.6.1 Initial value problems -- 7.6.2 Existence and uniqueness for nonlinear systems -- 7.6.3 Linear boundary value problems -- Chapter 7 Review -- Appendix A: Review Of Partial Fraction Expansions -- Appendix B: Review Of Determinants -- Appendix C: Review Of Gauss Elimination -- Appendix D: Review Of Complex Numbers And The Complex Plane -- Answers To Exercises -- Index -- Selected formulas.
9781118243381
Electronic books.
Ordinary Differential Equations. - 1 online resource (549 pages) - New York Academy of Sciences Ser. . - New York Academy of Sciences Ser. .
Cover -- Title Page -- Copyright -- Contents -- Preface -- 1 First-Order Differential Equations -- 1.1 Motivation And Overview -- 1.1.1 Introduction -- 1.1.2 Modeling -- 1.1.3 The order of a differential equation -- 1.1.4 Linear and nonlinear equations -- 1.1.5 Ourplan -- 1.1.6 Direction field -- 1.1.7 Computer software -- 1.2 Linear First-Order Equations -- 1.2.1 The simplest case -- 1.2.2 The homogeneous equation -- 1.2.3 Solving the full equation by the integrating factor method -- 1.2.4 Existence and uniqueness for the linear equation -- 1.3 Applications Of Linear First-Order Equations -- 1.3.1 Population dynamics -- exponential model -- 1.3.2 Radioactive decay -- carbon dating -- 1.3.3 Mixing problems -- a one-compartment model -- 1.3.4 The phase line, equilibrium points, and stability -- 1.3.5 Electrical circuits -- 1.4 Nonlinear First-Order Equations That Are Separable -- 1.5 Existence And Uniqueness -- 1.5.1 An existence and uniqueness theorem -- 1.5.2 Illustrating the theorem -- 1.5.3 Application to free fall -- physical significance of nonuniqueness -- 1.6 Applications Of Nonlinear First-Order Equations -- 1.6.1 The logistic model of population dynamics -- 1.6.2 Stability of equilibrium points and linearized stability analysis -- 1.7 Exact Equations And Equations That Can Be Made Exact -- 1.7.1 Exact differential equations -- 1.7.2 Making an equation exact -- integrating factors -- 1.8 Solution By Substitution -- 1.8.1 Bernoulli's equation -- 1.8.2 Homogeneous equations -- 1.9 Numerical Solution By Euler's Method -- 1.9.1 Euler's method -- 1.9.2 Convergence of Euler's method -- 1.9.3 Higher-order methods -- Chapter 1 Review -- 2 Higher-Order Linear Equations -- 2.1 Linear Differential Equations Of Second Order -- 2.1.1 Introduction -- 2.1.2 Operator notation and linear differential operators -- 2.1.3 Superposition principle. 2.2 Constant-Coefficient Equations -- 2.2.1 Constant coefficients -- 2.2.2 Seeking a general solution -- 2.2.3 Initial value problem -- 2.3 Complex Roots -- 2.3.1 Complex exponential function -- 2.3.2 Complex characteristic roots -- 2.4 Linear Independence -- Existence, Uniqueness, General Solution -- 2.4.1 Linear dependence and linear independence -- 2.4.2 Existence, uniqueness, and general solution -- 2.4.3 Abel's formula and Wronskian test for linear independence -- 2.4.4 Building a solution method on these results -- 2.5 Reduction Of Order -- 2.5.1 Deriving the formula -- 2.5.2 The method rather than the formula -- 2.5.3 About the method of reduction of order -- 2.6 Cauchy-Euler Equations -- 2.6.1 General solution -- 2.6.2 Repeated roots and reduction of order -- 2.6.3 Complex roots -- 2.7 The General Theory For Higher-Order Equations -- 2.7.1 Theorems for nth-order linear equations -- 2.7.2 Constant-coefficient equations -- 2.7.3 Cauchy-Euler equations -- 2.8 Nonhomogeneous Equations -- 2.8.1 General solution -- 2.8.2 The scaling and superposition of forcing functions -- 2.9 Particular Solution By Undetermined Coefficients -- 2.9.1 Undetermined coefficients -- 2.9.2 A special case -- the complex exponential method -- 2.10 Particular Solution By Variation Of Parameters -- 2.10.1 First-order equations -- 2.10.2 Second-order equations -- Chapter 2 Review -- 3 Applications Of Higher-Order Equations -- 3.1 Introduction -- 3.2 Linear Harmonic Oscillator -- Free Oscillation -- 3.2.1 Mass-spring oscillator -- 3.2.2 Undamped free oscillation -- 3.2.3 Pendulum -- 3.3 Free Oscillation With Damping -- 3.3.1 Underdamped -- 3.3.2 Critically damped -- 3.3.3 Overdamped -- 3.4 Forced Oscillation -- 3.4.1 Undamped, c = 0 -- 3.4.2 Damped, c > -- 0 -- 3.5 Steady-State Diffusion -- A Boundary Value Problem -- 3.5.1 Boundary value problems. existence and uniqueness -- 3.5.2 Steady-state heat conduction in a rod -- 3.6 Introduction To The Eigenvalue Problem -- Column Buckling -- 3.6.1 An eigenvalue problem -- 3.6.2 Application to column buckling -- Chapter 3 Review -- 4 Systems Of Linear Differential Equations -- 4.1 Introduction, And Solution By Elimination -- 4.1.1 Introduction -- 4.1.2 Physical examples -- 4.1.3 Solutions, existence, and uniqueness -- 4.1.4 Solution by elimination -- 4.1.5 Auxiliary variables -- 4.2 Application To Coupled Oscillators -- 4.2.1 Coupled oscillators -- 4.2.2 Reduction to first-order system by auxiliary variables -- 4.2.3 The free vibration -- 4.2.4 The forced vibration -- 4.3 N-Space And Matrices -- 4.3.1 Passage from 2-space to n-space -- 4.3.2 Matrix operators on vectors in n-space -- 4.3.3 Identity matrix and zero matrix -- 4.3.4 Relevance to systems of linear algebraic equations -- 4.3.5 Vector and matrix functions -- 4.4 Linear Dependence And Independence Of Vectors -- 4.4.1 Linear dependence of a set of constant vectors in n-space -- 4.4.2 Linear dependence of vector functions in n-space -- 4.5 Existence, Uniqueness, And General Solution -- 4.5.1 The key theorems -- 4.5.2 Illustrating the theorems -- 4.6 Matrix Eigenvalue Problem -- 4.6.1 The eigenvalue problem -- 4.6.2 Solving an eigenvalue problem -- 4.6.3 Complex eigenvalues and eigenvectors -- 4.7 Homogeneous Systems With Constant Coefficients -- 4.7.1 Solution by the method of assumed exponential form -- 4.7.2 Application to the two-mass oscillator -- 4.7.3 The case of repeated eigenvalues -- 4.7.4 Modifying the method if there are defective eigenvalues -- 4.7.5 Complex eigenvalues -- 4.8 Dot Product And Additional Matrix Algebra -- 4.8.1 More about n-space: dot product, norm, and angle -- 4.8.2 Algebra of matrix operators -- 4.8.3 Inverse matrix. 4.9 Explicit Solution Of x' = Ax And The Matrix Exponential Function -- 4.9.1 Matrix exponential solution -- 4.9.2 Getting the exponential matrix series into closed form -- 4.10 Nonhomogeneous Systems -- 4.10.1 Solution by variation of parameters -- 4.10.2 Constant coefficient matrix -- 4.10.3 Particular solution by undetermined coefficients -- Chapter 4 Review -- 5 Laplace Transform -- 5.1 Introduction -- 5.2 The Transform And Its Inverse -- 5.2.1 Laplace transform -- 5.2.2 Linearity property of the transform -- 5.2.3 Exponential order, piecewise continuity, and conditions for existence of the transform -- 5.2.4 Inverse transform -- 5.2.5 Introduction to the determination of inverse transforms -- 5.3 Application To The Solution Of Differential Equations -- 5.3.1 First-order equations -- 5.3.2 Higher-order equations -- 5.3.3 Systems -- 5.3.4 Application to a nonconstant-coefficient equation -- Bessel's equation -- 5.4 Discontinuous Forcing Functions -- Heaviside Step Function -- 5.4.1 Motivation -- 5.4.2 Heaviside step function and piecewise-defined functions -- 5.4.3 Transforms of Heaviside and time-delayed functions -- 5.4.4 Differential equations with piecewise-defined forcing functions -- 5.4.5 Periodic forcing functions -- 5.5 Convolution -- 5.5.1 Definition of Laplace convolution -- 5.5.2 Convolution theorem -- 5.5.3 Applications -- 5.5.4 Integro-differential equations and integral equations -- 5.6 Impulsive Forcing Functions -- Dirac Delta Function -- 5.6.1 Impulsive forces -- 5.6.2 Dirac delta function -- 5.6.3 The jump caused by the delta function -- 5.6.4 Caution -- 5.6.5 Impulse response function -- Chapter 5 Review -- 6 Series Solutions -- 6.1 Introduction -- 6.2 Power Series And Taylor Series -- 6.2.1 Power series -- 6.2.2 Manipulation of power series -- 6.2.3 Taylor series -- 6.3 Power Series Solution About A Regular Point. 6.3.1 Power series solution theorem -- 6.3.2 Applications -- 6.4 Legendre And Bessel Equations -- 6.4.1 Introduction -- 6.4.2 Legendre's equation -- 6.4.3 Bessel's equation -- 6.5 The Method of Frobenius -- 6.5.1 Motivation -- 6.5.2 Regular and irregular singular points -- 6.5.3 The method of Frobenius -- Chapter 6 Review -- 7 Systems Of Nonlinear Differential Equations -- 7.1 Introduction -- 7.2 The Phase Plane -- 7.2.1 Phase plane method -- 7.2.2 Application to nonlinear pendulum -- 7.2.3 Singular points and their stability -- 7.3 Linear Systems -- 7.3.1 Introduction -- 7.3.2 Purely imaginary eigenvalues (Center) -- 7.3.3 Complex conjugate eigenvalues (Spiral) -- 7.3.4 Real eigenvalues of the same sign (Node) -- 7.3.5 Real eigenvalues of opposite sign (Saddle) -- 7.4 Nonlinear Systems -- 7.4.1 Local linearization -- 7.4.2 Predator-prey population dynamics -- 7.4.3 Competing species -- 7.5 Limit Cycles -- 7.6 Numerical Solution Of Systems By Euler's Method -- 7.6.1 Initial value problems -- 7.6.2 Existence and uniqueness for nonlinear systems -- 7.6.3 Linear boundary value problems -- Chapter 7 Review -- Appendix A: Review Of Partial Fraction Expansions -- Appendix B: Review Of Determinants -- Appendix C: Review Of Gauss Elimination -- Appendix D: Review Of Complex Numbers And The Complex Plane -- Answers To Exercises -- Index -- Selected formulas.
9781118243381
Electronic books.