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

Download PDF For Free: Differential Equations With Applications and Historical Notes Third Edition by George F. Simmons

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