Differential Equations With Applications and Historical Notes

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Differential Equations With Applications and Historical Notes

PDF Free Download | Differential Equations With Applications and Historical Notes 3rd Edition by George F. Simmons

Contents of Differential Equations PDF

  • The Nature of Differential Equations Separable Equations
  • Introduction
  • General Remarks on Solutions
  • Families of Curves Orthogonal Trajectories
  • Growth, Decay, Chemical Reactions, and Mixing
  • Falling Bodies and Other Motion Problems
  • The Brachistochrone Fermat and the Bernoullis
  • Appendix A: Some Ideas From the Theory of Probability:
  • The Normal Distribution Curve (or Bell Curve) and
  • Its Differential Equation
  • First Order Equations
  • Homogeneous Equations
  • Exact Equations
  • Integrating Factors
  • Linear Equations
  • Reduction of Order
  • The Hanging Chain Pursuit Curves
  • Simple Electric Circuits
  • Second Order Linear Equations
  • Introduction
  • The General Solution of the Homogeneous Equation
  • The Use of a Known Solution to find Another
  • The Homogeneous Equation with Constant Coefficients
  • The Method of Undetermined Coefficients
  • The Method of Variation of Parameters
  • Vibrations in Mechanical and Electrical Systems
  • Newton’s Law of Gravitation and The Motion of the Planets
  • Higher Order Linear Equations Coupled
  • Harmonic Oscillators
  • Operator Methods for Finding Particular Solutions
  • Appendix A Euler
  • Appendix B Newton
  • Qualitative Properties of Solutions
  • Oscillations and the Sturm Separation Theorem
  • The Sturm Comparison Theorem
  • Power Series Solutions and Special Functions
  • Introduction A Review of Power Series
  • Series Solutions of First Order Equations
  • Second Order Linear Equations Ordinary Points
  • Regular Singular Points
  • Regular Singular Points (Continued)
  • Gauss’s Hypergeometric Equation
  • The Point at Infinity
  • Appendix A Two Convergence Proofs
  • Appendix B Hermite Polynomials and Quantum Mechanics
  • Appendix C Gauss
  • Appendix D Chebyshev Polynomials and the Minimax Property
  • Appendix E Riemann’s Equation
  • Fourier Series and Orthogonal Functions
  • The Fourier Coefficients
  • The Problem of Convergence
  • Even and Odd Functions Cosine and Sine Series
  • Extension to Arbitrary Intervals
  • Orthogonal Functions
  • The Mean Convergence of Fourier Series
  • Appendix A A Pointwise Convergence Theorem
  • Partial Differential Equations and Boundary Value Problems
  • Introduction Historical Remarks
  • Eigenvalues, Eigenfunctions, and the Vibrating String
  • The Heat Equation
  • The Dirichlet Problem for a Circle Poisson’s Integral
  • Sturm–Liouville Problems
  • Appendix A The Existence of Eigenvalues and Eigenfunctions
  • Some Special Functions of Mathematical Physics
  • Legendre Polynomials
  • Properties of Legendre Polynomials
  • Bessel Functions The Gamma Function
  • Properties of Bessel Functions
  • Appendix A Legendre Polynomials and Potential Theory
  • Appendix B Bessel Functions and the Vibrating Membrane
  • Appendix C Additional Properties of Bessel Functions
  • Laplace Transforms
  • Introduction
  • A Few Remarks on the Theory
  • Applications to Differential Equations
  • Derivatives and Integrals of Laplace Transforms
  • Convolutions and Abel’s Mechanical Problem
  • More about Convolutions The Unit Step and
  • Impulse Functions
  • Appendix A Laplace
  • Appendix B Abel
  • Systems of First Order Equations
  • General Remarks on Systems
  • Linear Systems
  • Homogeneous Linear Systems with Constant Coefficients
  • Nonlinear Systems Volterra’s Prey-Predator Equations
  • Nonlinear Equations
  • Autonomous Systems The Phase Plane and Its Phenomena
  • Types of Critical Points Stability
  • Critical Points and Stability for Linear Systems
  • Stability By Liapunov’s Direct Method
  • Simple Critical Points of Nonlinear Systems
  • Nonlinear Mechanics Conservative Systems
  • Periodic Solutions The Poincaré–Bendixson Theorem
  • More about the van der Pol Equation
  • Appendix A Poincaré
  • Appendix B Proof of Liénard’s Theorem
  • The Calculus of Variations
  • Introduction Some Typical Problems of the Subject
  • Euler’s Differential Equation for an Extremal
  • Isoperimetric Problems
  • Appendix A Lagrange
  • Appendix B Hamilton’s Principle and Its Implications
  • The Existence and Uniqueness of Solutions
  • The Method of Successive Approximations
  • Picard’s Theorem
  • Systems The Second Order Linear Equation
  • Numerical Methods
  • By John S Robertson
  • Introduction
  • The Method of Euler
  • Errors
  • An Improvement to Euler
  • Higher Order Methods The Nature of Differential Equations Separable Equations
  • Introduction
  • General Remarks on Solutions
  • Families of Curves Orthogonal Trajectories
  • Growth, Decay, Chemical Reactions, and Mixing
  • Falling Bodies and Other Motion Problems
  • The Brachistochrone Fermat and the Bernoullis
  • Appendix A: Some Ideas From the Theory of Probability:
  • The Normal Distribution Curve (or Bell Curve) and
  • Its Differential Equation
  • First Order Equations
  • Homogeneous Equations
  • Exact Equations
  • Integrating Factors
  • Linear Equations
  • Reduction of Order
  • The Hanging Chain Pursuit Curves
  • Simple Electric Circuits
  • Second Order Linear Equations
  • Introduction
  • The General Solution of the Homogeneous Equation
  • The Use of a Known Solution to find Another
  • The Homogeneous Equation with Constant Coefficients
  • The Method of Undetermined Coefficients
  • The Method of Variation of Parameters
  • Vibrations in Mechanical and Electrical Systems
  • Newton’s Law of Gravitation and The Motion of the Planets
  • Higher Order Linear Equations Coupled
  • Harmonic Oscillators
  • Operator Methods for Finding Particular Solutions
  • Appendix A Euler
  • Appendix B Newton
  • Qualitative Properties of Solutions
  • Oscillations and the Sturm Separation Theorem
  • The Sturm Comparison Theorem
  • Power Series Solutions and Special Functions
  • Introduction A Review of Power Series
  • Series Solutions of First Order Equations
  • Second Order Linear Equations Ordinary Points
  • Regular Singular Points
  • Regular Singular Points (Continued)
  • Gauss’s Hypergeometric Equation
  • The Point at Infinity
  • Appendix A Two Convergence Proofs
  • Appendix B Hermite Polynomials and Quantum Mechanics
  • Appendix C Gauss
  • Appendix D Chebyshev Polynomials and the Minimax Property
  • Appendix E Riemann’s Equation
  • Fourier Series and Orthogonal Functions
  • The Fourier Coefficients
  • The Problem of Convergence
  • Even and Odd Functions Cosine and Sine Series
  • Extension to Arbitrary Intervals
  • Orthogonal Functions
  • The Mean Convergence of Fourier Series
  • Appendix A A Pointwise Convergence Theorem
  • Partial Differential Equations and Boundary Value Problems
  • Introduction Historical Remarks
  • Eigenvalues, Eigenfunctions, and the Vibrating String
  • The Heat Equation
  • The Dirichlet Problem for a Circle Poisson’s Integral
  • Sturm–Liouville Problems
  • Appendix A The Existence of Eigenvalues and Eigenfunctions
  • Some Special Functions of Mathematical Physics
  • Legendre Polynomials
  • Properties of Legendre Polynomials
  • Bessel Functions The Gamma Function
  • Properties of Bessel Functions
  • Appendix A Legendre Polynomials and Potential Theory
  • Appendix B Bessel Functions and the Vibrating Membrane
  • Appendix C Additional Properties of Bessel Functions
  • Laplace Transforms
  • Introduction
  • A Few Remarks on the Theory
  • Applications to Differential Equations
  • Derivatives and Integrals of Laplace Transforms
  • Convolutions and Abel’s Mechanical Problem
  • More about Convolutions The Unit Step and
  • Impulse Functions
  • Appendix A Laplace
  • Appendix B Abel
  • Systems of First Order Equations
  • General Remarks on Systems
  • Linear Systems
  • Homogeneous Linear Systems with Constant Coefficients
  • Nonlinear Systems Volterra’s Prey-Predator Equations
  • Nonlinear Equations
  • Autonomous Systems The Phase Plane and Its Phenomena
  • Types of Critical Points Stability
  • Critical Points and Stability for Linear Systems
  • Stability By Liapunov’s Direct Method
  • Simple Critical Points of Nonlinear Systems
  • Nonlinear Mechanics Conservative Systems
  • Periodic Solutions The Poincaré–Bendixson Theorem
  • More about the van der Pol Equation
  • Appendix A Poincaré
  • Appendix B Proof of Liénard’s Theorem
  • The Calculus of Variations
  • Introduction Some Typical Problems of the Subject
  • Euler’s Differential Equation for an Extremal
  • Isoperimetric Problems
  • Appendix A Lagrange
  • Appendix B Hamilton’s Principle and Its Implications
  • The Existence and Uniqueness of Solutions
  • The Method of Successive Approximations
  • Picard’s Theorem
  • Systems The Second Order Linear Equation
  • Numerical Methods
  • By John S Robertson
  • Introduction
  • The Method of Euler
  • Errors
  • An Improvement to Euler
  • Higher Order Methods
  • Systems

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