What are you going to learn?

Model with first-order differential equations (DE) and identify initial value problems. Solve scalar differential equations, homogeneous and non-homogeneous, using methods including separation of variables, integrating factors, eigenvalues, LaPlace transformations. Model with systems of first-order DEs and higher-order DEs. Solve systems of linear differential equations using matrices and eigenvalues.

Content

Chapter 1. Introduction

  • Terminology
  • Classification of Differential Equations
  • Initial Value Problem. Cauchy’s Theorem of y’ = f(x, y)
  • Mathematical Models and Direction of Fields

Chapter 2. First Order Ordinary Differential Equations (ODE)

  • Integrating Factors
  • Separation of Variables
  • Exact Equations
  • First Order Linear Equations
  • Equations with Homogeneous Coefficients
  • Solution by Substitutions
  • Bernoulli’s DE
  • Computer Solutions

Chapter 3. Second Order Ordinary Differential Equations

  • The General Second Order Linear Homogeneous Equation
  • Linear Independence
  • Existence and Uniqness Theorem
  • The Wronskian
  • General Solution of a Homogeneous Equation
  • General Solution of a Nonhomogeneous Equation
  • Differential Operators

Chapter 4. Higher Order Constant Coefficients Linear Equations

  • Introduction
  • Reduction of Order
  • Homogeneous LE with Constant Coefficients
  • Undetermined Coefficients – Superposition Approach
  • Variation of Parameters
  • Cauchy – Euler Equation
  • Systems of Linear Equations
  • Nonlinear Equations

Chapter 5. Series Solutions of 2d Order LDE

  • Power Series
  • Series Solutions Near an Ordinary Point
  • Euler Equations, Regular Singular Points
  • Series Solutions Near an Regular Singular Point
  • Bessel’s Equation

Chapter 6. Laplace Transform

  • Definition
  • Solution of Initial Value Probems
  • Step Fucntions
  • Impulse Functions
  • The Convolution Integral

Chapter 7. Systems of First Order Differential Equations

  • Review of Matrices
  • Systems of Linear Equations. Linear Independence
  • Eigenvectors, Eigenvalues
  • Basic Theory
  • Homogeneous Linear Systems with Constant Coefficients
  • Nonhomogeneous Linear Systems

Chapter 8. Numerical Methods

  • The Euler’s Method
  • Runge-Kutta Method
  • Multi Step Methods
  • Systems of First order Equations

Bibliography

  • Boyce, W. E., DiPrima, R. Elementary Differential Equations and Boundary Problems. 8th ed., Wiley & Sons Inc., New York.
  • Zill, D. A First Course in Differential Equations with Modeling Applications. 10th ed., Brooks/Cole, London, 1997.

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