MATH-3266: Differential Equations I

• Course Description
  
  
  

  I. LECTURES

  
  

  STATEMENT: Most of lectures in this course are delivered by blackboard writing, students are required to take lecture notes on class, but we use power point slides for some sections or some portions of sections having many notions and long formulas. Sections labelled with * are reading materials or will be lectured in the course Differential Equations II.

  
  

  PART I: INTRODUCTION

  Chapter 1: Introduction

  1.1. Mathematical Models Described by Differential Equtations and Geometric Meaning of Differential Equations -- Direction Fields
  
  1.2. Solutions of Some Differential Equations: Features and Geometric Meanings of Solutions
  
  1.3. Classification of Differential Equations and Relevant Notions
  
  

  * 1.4. Historical Remarks (Reading Material)

  
  

  PART II: BASIC MISCELLANEOUS METHODS OF SOLVING LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELLING

  Chapter 2: First Order Differential Equations

  1. Solve First Order LINEAR Differential Equations

  2.1. Method of Integrating Factors for Solving First Order LINEAR Equations

  2. Solve First Order NON-LINEAR Equation

  2.2. Solve Separable Non-Linear Differential Equations and Equations That Can Be Converted Into Separable Equations

  2.6. Sovle Exact Non-Linar Differential Equations and Equations Convertible Into an Exact Equation by Integrating Factors

  2.4. Sufficient Conditions for Existence and Uniquness of First-Order Differential Euqations and Difference between Linear and Nonlinear Equations
  
  
  3. Numerical Approximation Method of Solving Differential Equations

  *2.7. Numerical Approximations Using Euler's Method (or Tangent Line Method) (to be lectured in Chapter 8)

  4. Mathematical Theory on Solutions of Differential Equations: Existence and Uniquness
  
  2.8. The Existence and Uniqueness of the Solution of Nonlinear Equations Studied by Iteration Method
  
  
  5. Application of First Order Differential Equations in Natural Science, Social Science and Engineeing
  
  2.3. Modeling with First Order Equations (Quick Review)
  

  * 2.5. Autonomous Equations and Population Dynamics (Reading Material)

  6. Difference Equations: Discrete Counterparts of Differential Equations

  * 2.9. First Order Difference Equations (Reading Material)

  
  

  Chapter 3: Second Order Linear Equations

  1. Solve Homogeneous Differential Equations with Constant Coefficients Using Method of Characteristic Equation
  
  3.1. Homogeneous Equation with Constant Coefficients and The Solution in the Case of Characteristic Equation Having Two Real Unequal Roots
  
  3.2. Mathematical Theory on Solutions of Linear Homogeneous Equations: Non-zero Wronskian Determinant as the Sufficient and Necessary Condition for the Solution Expressed as Linear Superposition of Two Fundamental Solutions
  
  3.3. Solution of A Second Order Linear Homgeneous Equation with Constant Coeffcients Whose Characteristic Equation Has Two Conjugate Complex Roots
  
  3.4. Solve A Second Order Linear Homgeneous Equation with Constant Coeffcients Whose Characteristic Equation Has Two Repeated Roots by the Method of Reduction of Order
  
  
  2. Two Methods of Finding A Particular Solution of A Non-homogeneous Second Order Differential Equation with Constant Coefficients
  
  3.5. Method of Undertmined Coefficients
  
  3.6. Method of Parameter Variation
  
  
  3. Application of Second-Order Linear Differential Equation in Physics: Vibrations in Classical Mechanics and LCR Circuit in Electromagnetism

  3.7 Damped and Undamped Free Vibrations in Mechanics and LCR Circuit in Electromagnetism (Reading Material)

  3.8 Forced Damped and Undamped Vibrations in Mechanics and Relevant Physical Phenomena: Resonance and Beat (Reading Material)

  
  

  Chapter 4: Higher Order Linear Equations

  1. General Theory
  
  4.1. General Theory of nth Order Linear Equations: General Form, Homogeneous and Non-homogeneous, Linear Independent Fundamental Solutions, Particular Solution
  
  
  2. Solve Homogeneous Equations
  
  4.2. Solve A Homogeneous Equation with Constant Coefficients in the Cases that the Characteristic Equation has Real Unequal Roots, Complex Unequal Roots, Real or Complex Repeated Roots
  
  
  3. Solve Non-Homogeneous Equations
  
  4.3. Method of Undetermined Coefficients
  
  4.4. Method of Variation of Parameters
  
  
  
  

  PART III: POWER SERIES SOLUTION TO SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF VARIABLE COEFFICIENTS

  Chapter 5: Series Solutions of Second Order Linear Equations

  5.1. Review on Some Aspects of Power Series: Convergence, Radius and Interval of Convergence, Test of Convergence; Shift of Index of Summation in Power Series (Quick Review)
  
  5.2. Part I of Series Solution near An Ordinary Point: Illustration on Features of Series Solution near An Ordinary Point with Examples
  
  5.3. Part II of Series Solution near An Ordinary Point: General Theory
  
  5.4. Euler Equation as An Illustrating Example for Differential Equations with Regular Singular Point: Features of Series Solution Near A Regular Singular Point
  
  5.5. Part I of Series Solution near A Regular Singular Point: Illustration on Features of Series Solution near A Regular Singular Point with Examples
  
  5.6. Part II of Series Solution near A Regular Singular Point: General Theory
  
  5.7. A Example of Second Order Differential Equation with Coefficients Having Regular Singular Point: Bessel's Equation
  
  
  
  
  

  II. WEEKLY HOMEWORK ASSIGNMENTS

  
  
  1. Homework Assignment 1
     
     
  2. Homework Assignment 2
     
     
  3. Homework Assignment 3
     
     
  4. Homework Assignment 4
     
     
  5. Homework Assignment 5
     
     
  6. Homework Assignment 6
     
     
  7. Homework Assignment 7
     
     
  8. Homework Assignment 8
     
     
  9. Homework Assignment 9
     
     
  10. Homework Assignment 10
      
      
  11. Homework Assignment 11
      
      
  12. Homework Assignment 12