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ODEs and PDEs

Differential equations are divided into two main types: Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs).

Ordinary Differential Equations (ODEs)​

ODEs involve derivatives with respect to only one independent variable. They describe systems with a single dimension, like motion over time (e.g., a harmonic oscillator describing spring motion):

dfdt=g(f,t)\frac{\mathrm{d}f}{\mathrm{d}t} = g(f, t)

Partial Differential Equations (PDEs)​

PDEs involve derivatives with respect to multiple independent variables. They describe more complex systems evolving over space and time (e.g., heat diffusion), and are used in fields like fluid dynamics, electromagnetism, and quantum mechanics:

∂u∂t=Δu=∂2u∂x2+∂2u∂y2\frac{\partial u}{\partial t} = \Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}