Monday, September 5, 2011

Differential Equations Demystified

Differential Equations Demystified
| Pages | PDF | 4.9 MB |


What Is a Differential Equation? 
  • Introductory Remarks 
  • The Nature of Solutions 
  • Separable Equations 
  • First-Order Linear Equations 
  • Exact Equations 
  • Orthogonal Trajectories and Families of Curves 
  • Homogeneous Equations 
  • Integrating Factors 
  • Reduction of Order
  • The Hanging Chain and Pursuit Curves 
  • Electrical Circuits 
  • Exercises

Second-Order Equations 
  • Second-Order Linear Equations with Constant Coefficients 
  • The Method of Undetermined Coefficients 
  • The Method of Variation of Parameters 
  • The Use of a Known Solution to Find Another 
  • Vibrations and Oscillations 
  • Newton’s Law of Gravitation and Kepler’s Laws 
  • Higher-Order Linear Equations,
  • Coupled Harmonic Oscillators 
  • Exercises

Power Series Solutions and Special Functions 
  • Introduction and Review of Power Series 
  • Series Solutions of First-Order Differential Equations 
  • Second-Order Linear Equations Ordinary Points 
  • Exercises

Fourier Series: Basic Concepts 
  • Fourier Coefficients 
  • Some Remarks About Convergence 
  • Even and Odd Functions: Cosine and Sine Series 
  • Fourier Series on Arbitrary Intervals 
  • Orthogonal Functions 
  • Exercises

Partial Differential Equations and Boundary Value Problems 
  • Introduction and Historical Remarks 
  • Eigenvalues, Eigenfunctions, and the Vibrating String 
  • The Heat Equation: Fourier’s Point of View 
  • The Dirichlet Problem for a Disc 
  • Sturm–Liouville Problems 
  • Exercises 

Laplace Transforms 
  • Introduction 
  • Applications to Differential Equations 
  • Derivatives and Integrals of Laplace Transforms 
  • Convolutions 
  • The Unit Step and Impulse Functions 
  • Exercises

Numerical Methods 
  • Introductory Remarks 
  • The Method of Euler 
  • The Error Term 
  • An Improved Euler Method 
  • The Runge–Kutta Method 
  • Exercises

Systems of First-Order Equations 
  • Introductory Remarks 
  • Linear Systems 
  • Homogeneous Linear Systems with Constant Coefficients 
  • Nonlinear Systems: Volterra’s Predator–Prey Equations 
  • Exercises
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