Differential Equations

          Module 1: Fundamentals of Differential Equations

Section 1: Introduction to Differential Equations
  Lesson 1: What is a Differential Equation?  
    Module 1, Section 1, Lesson 1: An overview of what differential equations are and why they matter.
  Lesson 2: History and Motivation  
    Module 1, Section 1, Lesson 2: A look at the origins of differential equations and their role in modeling change.

Section 2: Classification and Key Definitions
  Lesson 1: Order and Degree of Differential Equations  
    Module 1, Section 2, Lesson 1: Understanding the terminology of order, degree, and classification.
  Lesson 2: Ordinary versus Partial Differential Equations  
    Module 1, Section 2, Lesson 2: Introducing the differences between ordinary and partial differential equations.

Section 3: Fundamental Concepts and Terminology
  Lesson 1: General and Particular Solutions  
    Module 1, Section 3, Lesson 1: Defining what constitutes general and particular solutions in differential equations.
  Lesson 2: Initial and Boundary Conditions  
    Module 1, Section 3, Lesson 2: Explaining the importance of conditions for uniquely determining a solution.

Section 4: Graphical Interpretations and Basic Properties
  Lesson 1: Direction and Slope Fields  
    Module 1, Section 4, Lesson 1: Learning how to visualize solutions using direction fields.
  Lesson 2: Equilibrium Solutions and Stability  
    Module 1, Section 4, Lesson 2: Exploring equilibrium solutions and their stability in simple systems.

Section 5: Review and Fundamental Problem Solving
  Lesson 1: Basic Examples and Problem Sets  
    Module 1, Section 5, Lesson 1: Working through simple differential equation examples.
  Lesson 2: Recap of Key Concepts  
    Module 1, Section 5, Lesson 2: Reviewing and reinforcing the fundamental ideas presented in Module 1.

Module 2: First Order Differential Equations

Section 1: Separable Equations
  Lesson 1: Recognizing Separable Equations  
    Module 2, Section 1, Lesson 1: Identifying differential equations that can be separated into functions of x and y.
  Lesson 2: Step-by-Step Solution of Separable Equations  
    Module 2, Section 1, Lesson 2: Solving separable equations using integration.

Section 2: Linear Differential Equations
  Lesson 1: Characteristics of Linear Equations  
    Module 2, Section 2, Lesson 1: Recognizing the form and properties of first order linear differential equations.
  Lesson 2: The Integrating Factor Method  
    Module 2, Section 2, Lesson 2: Learning to solve linear equations by applying the integrating factor.

Section 3: Exact Differential Equations
  Lesson 1: Checking for Exactness  
    Module 2, Section 3, Lesson 1: Understanding the test and conditions for an equation to be exact.
  Lesson 2: Solving Exact Equations  
    Module 2, Section 3, Lesson 2: Step-by-step techniques for solving exact differential equations.

Section 4: Non‑Exact Equations and Finding Integrating Factors
  Lesson 1: Converting Non‑Exact to Exact Equations  
    Module 2, Section 4, Lesson 1: Learning methods for determining integrating factors.
  Lesson 2: Application Examples Using Integrating Factors  
    Module 2, Section 4, Lesson 2: Practicing with examples that require integrating factors to solve.

Section 5: Applications of First Order DEs
  Lesson 1: Modeling Natural and Social Phenomena  
    Module 2, Section 5, Lesson 1: Applying first order differential equations to model real‑world scenarios.
  Lesson 2: Interpretation and Analysis of Solutions  
    Module 2, Section 5, Lesson 2: Discussing how solutions relate to physical and societal phenomena.

Module 3: Second Order Differential Equations

Section 1: Introduction to Second Order Differential Equations
  Lesson 1: Structure and Examples of Second Order DEs  
    Module 3, Section 1, Lesson 1: Defining second order differential equations and reviewing simple examples.
  Lesson 2: Applications and Relevance  
    Module 3, Section 1, Lesson 2: Exploring why second order DEs are important in modeling real systems.

Section 2: Homogeneous Second Order Equations
  Lesson 1: The Characteristic Equation Method  
    Module 3, Section 2, Lesson 1: Learning to form and solve the characteristic equation.
  Lesson 2: Types of Roots and General Solutions  
    Module 3, Section 2, Lesson 2: Understanding real, repeated, and complex roots in the context of solutions.

Section 3: Non‑Homogeneous Second Order Equations
  Lesson 1: Introduction to Particular Solutions  
    Module 3, Section 3, Lesson 1: Explaining the concept of particular versus general solutions.
  Lesson 2: Method of Undetermined Coefficients  
    Module 3, Section 3, Lesson 2: A step‑by‑step approach to finding particular solutions for non‑homogeneous equations.

Section 4: Initial Value Problems for Second Order DEs
  Lesson 1: Setting Up Initial Conditions  
    Module 3, Section 4, Lesson 1: How to use initial conditions to determine unique solutions.
  Lesson 2: Solving IVPs in Second Order Equations  
    Module 3, Section 4, Lesson 2: Working through examples that involve initial value problems.

Section 5: Applications in Physics and Engineering
  Lesson 1: Modeling Oscillations and Vibrations  
    Module 3, Section 5, Lesson 1: Examining second order equations in mechanical and electrical oscillation.
  Lesson 2: Case Studies: Pendulums and Circuits  
    Module 3, Section 5, Lesson 2: Applying theory to model and analyze systems in physics and engineering.

Module 4: Methods and Techniques for Solving Differential Equations

Section 1: Overview of Analytical Methods
  Lesson 1: Traditional Analytical Techniques  
    Module 4, Section 1, Lesson 1: A review of classic methods used to solve differential equations.
  Lesson 2: Strengths and Limitations of Various Methods  
    Module 4, Section 1, Lesson 2: Discussing when and why to choose a particular solution technique.

Section 2: Introduction to Series Solutions
  Lesson 1: Power Series Methods in DEs  
    Module 4, Section 2, Lesson 1: Learning the basics of expressing solutions as power series.
  Lesson 2: Applying Series Solutions to Simple Equations  
    Module 4, Section 2, Lesson 2: Step‑by‑step examples demonstrating series solutions in practice.

Section 3: Laplace Transforms – The Basics
  Lesson 1: Understanding the Laplace Transform  
    Module 4, Section 3, Lesson 1: Introducing the concept and basic properties of Laplace transforms.
  Lesson 2: Solving Differential Equations Using Laplace Transforms  
    Module 4, Section 3, Lesson 2: Applying Laplace transform methods to basic differential equations.

Section 4: Introduction to Numerical Methods
  Lesson 1: Euler’s Method for Differential Equations  
    Module 4, Section 4, Lesson 1: Introducing Euler’s Method as a simple numerical technique.
  Lesson 2: Fundamentals of Other Numerical Approaches  
    Module 4, Section 4, Lesson 2: A brief look at more numerical methods without in‑depth complexity.

Section 5: Integrating Analytical and Applied Methods
  Lesson 1: Graphical Interpretation of Solutions  
    Module 4, Section 5, Lesson 1: Learning to connect analytical solutions with their graphical representations.
  Lesson 2: Combining Methods for Comprehensive Problem Solving  
    Module 4, Section 5, Lesson 2: Exploring how to integrate analytical and numerical methods for practical applications.

Module 5: Applications and Modeling with Differential Equations

Section 1: Differential Equations in Physics
  Lesson 1: Modeling Motion with Differential Equations  
    Module 5, Section 1, Lesson 1: Applying differential equations to problems in mechanics and motion.
  Lesson 2: Newton’s Laws and Differential Equations  
    Module 5, Section 1, Lesson 2: Connecting the principles of physics with differential equation models.

Section 2: Differential Equations in Engineering
  Lesson 1: Electrical Circuits and DEs  
    Module 5, Section 2, Lesson 1: Understanding how differential equations model electrical circuit behavior.
  Lesson 2: Mechanical Vibrations and System Modeling  
    Module 5, Section 2, Lesson 2: Applying differential equations to analyze mechanical vibrations.

Section 3: Differential Equations in Biology and Ecology
  Lesson 1: Population Dynamics and Logistic Models  
    Module 5, Section 3, Lesson 1: Using differential equations to model population growth and decline.
  Lesson 2: Basic Models in Epidemiology  
    Module 5, Section 3, Lesson 2: Exploring how differential equations can help understand the spread of diseases.

Section 4: Differential Equations in Economics
  Lesson 1: Modeling Economic Growth with DEs  
    Module 5, Section 4, Lesson 1: An introduction to applying differential equations in economic models.
  Lesson 2: Trends and Forecasting Through Differential Equations  
    Module 5, Section 4, Lesson 2: Learning how differential equations can assist in forecasting and analyzing trends.

Section 5: Capstone Projects and Course Review
  Lesson 1: Integrative Project: Applying Course Concepts  
    Module 5, Section 5, Lesson 1: Working on a comprehensive project that applies the fundamental techniques learned.
  Lesson 2: Final Review and Reflection  
    Module 5, Section 5, Lesson 2: Summarizing the key concepts and reflecting on practical applications of differential equations.