Differential Equations studies equations that relate functions to their derivatives. These equations describe how systems evolve over time and space.

Key Topics

• First-Order ODEs — Separable equations, linear equations, and exact equations.
• Second-Order Linear ODEs — Homogeneous and non-homogeneous equations, characteristic equation method.
• Systems of ODEs — Solving coupled equations using matrix methods and eigenvalues.
• Laplace Transforms — Converting differential equations into algebraic equations: \(\mathcal{L}\{f(t)\} = F(s)\).
• Series Solutions — Solving ODEs using power series methods.
• Partial Differential Equations — Heat equation, wave equation, and Laplace's equation.
• Fourier Series — Decomposing periodic functions into sines and cosines.

Why It Matters

Differential equations model virtually everything that changes: population dynamics, electrical circuits, planetary motion, heat transfer, financial markets, and chemical reactions. They are the primary language of the physical sciences.