Differential Equations

Differential equations are mathematical equations that relate functions with their derivatives. They are fundamental tools for modeling how quantities change over time or space, appearing throughout physics, engineering, biology, economics, and many other fields.

Types of Differential Equations

This section is organized into two main categories:

Ordinary Differential Equations (ODEs)

Equations involving functions of a single variable and their derivatives. These describe systems that evolve in one dimension (typically time).

Topics covered: - Basic concepts and terminology - First-order equations and solution methods - Applications of first-order equations - Higher-order linear equations - Laplace transforms and operator methods - Systems of differential equations - Series solutions - Boundary value problems

Partial Differential Equations (PDEs)

Equations involving functions of multiple variables and their partial derivatives. These describe phenomena that vary in multiple dimensions (such as heat diffusion in space and time).

Topics covered: - Fourier series - Heat equation - Wave equation - Laplace's equation - Separation of variables - Boundary conditions and initial value problems

Why Study Differential Equations?

Differential equations are the language of change. They allow us to: - Model physical phenomena (motion, heat, waves, electricity) - Predict future behavior of systems - Understand equilibrium states and stability - Optimize processes and control systems - Analyze biological populations and chemical reactions

Whether you're interested in theoretical mathematics or practical applications, differential equations provide powerful tools for understanding our dynamic world.