Differential Equations With Applications and Historical Notes by George F. Simmons
Book Details :
LanguageEnglish
Pages763
FormatPDF
Size3.77 MB


Differential Equations With Applications and Historical Notes by George F. Simmons



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Preface to Differential Equations eBook

  • The Nature of Differential Equations Separable Equations 
  • 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 
  • 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 
  • 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 
  • 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 
  • 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 
  • 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 
  • Some Special Functions of Mathematical Physics 
  • Legendre Polynomials 
  • Properties of Legendre Polynomials 
  • Bessel Functions The Gamma Function 
  • Properties of Bessel Functions 
  • Laplace Transforms 
  • 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 
  • 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 
  • The Calculus of Variations 
  • Introduction Some Typical Problems of the Subject 
  • Euler s Differential Equation for an Extremal 
  • Isoperimetric Problems 
  • The Existence and Uniqueness of Solutions 
  • The Method of Successive Approximations 
  • Picard s Theorem 
  • Systems The Second Order Linear Equation 
  • Numerical Methods 
  • The Method of Euler Errors 
  • An Improvement to Euler 
  • Higher Order Methods Systems