Differential Equations
Free Online Course • Introduction to Ordinary Differential Equations • Self-Paced
Welcome to Differential Equations!
This comprehensive course covers the fundamentals of ordinary differential equations (ODEs). You will learn to classify, solve, and apply differential equations to real-world problems in physics, engineering, biology, and economics. Each module includes detailed lessons, worked examples, practice problems, quizzes, and printable study materials.
Self-Paced
Learn on your schedule. All materials available 24/7.
Interactive
Worked examples with step-by-step solutions and instant-feedback practice.
Comprehensive
Study guides, quick reference cards, and detailed worked examples.
Research-Based
Designed using evidence-based learning principles.
Course Modules
Introduction to Differential Equations
Learn what differential equations are, explore direction fields, existence and uniqueness, and Euler's method.
- What Is a DE? Classification and Terminology
- Direction Fields and Solution Curves
- Existence and Uniqueness of Solutions
- Euler's Method for Numerical Approximation
First-Order ODEs I
Master separable, linear, exact equations, and substitution methods for first-order ODEs.
- Separable Equations
- First-Order Linear Equations and Integrating Factors
- Exact Equations
- Substitution Methods: Bernoulli and Homogeneous
Applications of First-Order ODEs
Apply first-order ODEs to growth, decay, population models, mixing, and cooling problems.
- Exponential Growth and Decay
- Population Models: Logistic Equation
- Mixing Problems
- Newton's Law of Cooling
Second-Order Linear ODEs
Solve homogeneous and nonhomogeneous second-order linear ODEs with constant coefficients.
- Homogeneous with Constant Coefficients
- Characteristic Equation: Real, Repeated, Complex Roots
- Undetermined Coefficients
- Variation of Parameters
Applications of Second-Order ODEs
Model vibrations, resonance, RLC circuits, and explore mechanical-electrical analogies.
- Spring-Mass: Free Vibrations
- Forced Vibrations and Resonance
- RLC Electrical Circuits
- Mechanical vs Electrical Analogies
Laplace Transforms
Use Laplace transforms to solve IVPs, handle discontinuous forcing, and apply partial fractions.
- Definition and Basic Transforms
- Inverse Laplace and Partial Fractions
- Solving IVPs with Laplace Transforms
- Step Functions and Discontinuous Forcing
Systems of Differential Equations
Solve systems using matrix methods, eigenvalues, and classify equilibria with phase portraits.
- Systems as Matrix Equations
- Eigenvalue Method: Real Distinct Eigenvalues
- Complex and Repeated Eigenvalues
- Phase Portraits and Stability
Series Solutions & Special Topics
Use power series to solve ODEs near ordinary and singular points, and preview boundary value problems.
- Power Series Review
- Series Solutions Near Ordinary Points
- Series Solutions Near Singular Points (Frobenius)
- Boundary Value Problems and Fourier Series Preview
Ready to Get Started?
Begin with Module 1 to build a strong foundation in differential equation concepts and terminology. All materials are completely free!
Learning Tips
- Review calculus prerequisites -- integration techniques are essential throughout
- Work through all lessons in order for best understanding
- Practice, practice, practice! Do all practice problems before the quiz
- Use the study guide to review before quizzes and exams
- Verify solutions by substituting back into the original equation
- Connect concepts -- many methods share underlying ideas
Prerequisites
This course assumes you have completed Calculus I and II (or equivalent). You should be comfortable with derivatives, integrals, integration techniques (substitution, integration by parts, partial fractions), and basic linear algebra concepts (matrices, determinants).