Unit 2945: Differential Equations

1. Introduction to Differential Equations

• Definition and basic concepts
  • What is a differential equation?
  • Dependent and independent variables
• Importance and applications in modeling real-world phenomena
• Classification of differential equations
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
  • Linear vs Non-linear differential equations

2. First Order Differential Equations

• Overview and significance
• Methods of solution:
  • Separable equations
    • Definition and method
    • Examples and practice problems
  • Exact equations
    • Condition for exactness
    • Solving exact equations
  • Integrating factor method
    • Derivation and application
  • Bernoulli equations
    • Form and solution technique

3. Second Order Linear Differential Equations

• General form and terminology
• Characteristic equation
  • Finding roots and their interpretation
• Homogeneous equations
  • Solution structure
  • Distinct real roots, repeated roots, and complex roots
• Non-homogeneous equations
  • Method of undetermined coefficients
  • Variation of parameters
  • Examples and applications

4. Laplace Transform and Differential Equations

• Introduction to Laplace transform
  • Definition and intuition
  • Common Laplace transforms
• Properties of Laplace transform
  • Linearity
  • Shifting theorems
  • Differentiation and integration in the Laplace domain
• Solving differential equations using Laplace transform
  • Procedure and step-by-step examples
  • Handling initial conditions

5. Systems of Differential Equations

• Introduction and motivation
• Matrix representation of systems
• Eigenvalues and eigenvectors
  • Calculation and significance
• Stability analysis of equilibrium points
• Methods for solving linear systems
  • Diagonalization method
  • Undetermined coefficients for systems

6. Numerical Methods for Differential Equations

• Need for numerical methods
• Euler's method
  • Algorithm and implementation
  • Error analysis
• Runge-Kutta methods
  • Fourth-order Runge-Kutta overview
  • Advantages over Euler's method
• Finite difference methods
  • Basic concepts
  • Application to boundary value problems

7. Applications of Differential Equations

• Physics
  • Motion, heat conduction, wave equations
• Engineering
  • Control systems, circuit analysis
• Biology
  • Population dynamics, epidemiology models
• Economics
  • Growth models, optimization
• Chemistry
  • Reaction kinetics
• Case studies demonstrating modeling and solution interpretation