Differential Equations

Equations relating functions to their derivatives, modeling how systems evolve over time and space.

• ODE (ordinary differential equation):: involves derivatives with respect to a single variable
• PDE (partial differential equation):: involves partial derivatives across multiple variables
• Navier-Stokes equations govern fluid dynamics; Maxwell’s equations govern electromagnetism
• Schrodinger equation governs quantum mechanics; Einstein field equations govern relativity
• existence and uniqueness theorems guarantee solutions under specified conditions
• dynamical systems and chaos theory study long-term behavior and sensitivity to initial conditions
• Solved via calculus, linear algebra, fourier transform, and numerical methods
• Related: optimization, probability, topology, wave, physics