Differential Equations Part I Basic Theory Course Syllabus

Overview: This course provides a structured introduction to the fundamental theory of ordinary differential equations (ODEs), designed for learners with a basic background in calculus. Over approximately 19 hours of content, participants will explore core concepts and solution techniques for first- and second-order differential equations, with emphasis on real-world applications in science and engineering. The course combines theory, practical problem-solving, and interactive exercises to build a strong foundation for further study or professional application.

Module 1: Introduction to Ordinary Differential Equations

Estimated time: 3 hours

• Basic definition and classification of differential equations
• Understanding ODEs and their role in modeling dynamic systems
• Key terminologies: order, linearity, and solutions
• Introduction to initial value problems

Module 2: First-Order Differential Equations

Estimated time: 4 hours

• Solution methods for separable differential equations
• Techniques for solving exact equations
• Integrating factors and their application
• Existence and uniqueness of solutions

Module 3: Linear Second-Order Differential Equations

Estimated time: 4 hours

• Introduction to linear second-order ODEs with constant coefficients
• Homogeneous equations and characteristic equations
• General solutions for distinct and repeated roots
• Method of undetermined coefficients

Module 4: Applications of Second-Order Differential Equations

Estimated time: 4 hours

• Modeling mechanical vibrations using mass-spring systems
• Analysis of damped and undamped oscillations
• Application to electrical circuits (RLC circuits)
• Interpreting physical behavior from mathematical solutions

Module 5: Final Project and Review

Estimated time: 4 hours

• Solving real-world problems using ODEs
• Hands-on modeling exercises in engineering contexts
• Comprehensive review of key concepts and solution techniques

Prerequisites

• Basic knowledge of single-variable calculus (derivatives and integrals)
• Familiarity with functions and algebraic manipulation
• Understanding of fundamental concepts in linear algebra (recommended but not required)

What You'll Be Able to Do After

• Understand and classify ordinary differential equations
• Solve first-order ODEs including separable and exact equations
• Analyze linear second-order differential equations with constant coefficients
• Apply ODEs to model real-world systems in physics and engineering
• Interpret solutions in the context of dynamic phenomena such as oscillations and circuits