Differential equations: ODEs vs. PDEs

In mathematics, we often hear about differential equations - equations that relate a function to its own rate of change. Within differential equations, you’ll find a distinction between Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs), but you may find yourself asking: What is the difference between the two?

Well, at its simplest (and it may sound silly to say), the difference between and ODE and a PDE is that an ODE is the number of independent variables.

Very simply, an ODE relates a function to its own rate of change via a single independent variable. Think velocity on a line, which is equal to the change in position over time: $v = \frac{dx}{dt}$

A PDE relates a function to its own rate of change via multiple independent variables using partial derivates. For this, think of the heat equation, which depends on location and time: $\frac{\partial u \left(x,t\right)}{\partial t} = \alpha \frac{\partial^2 u\left(x,t\right)}{\partial x ^2}$. In this case, $u\left(x,t\right)$ is a function of both x and t, therefore, we must take the partial derivative of the function u, meaning we hold the other independent variables as constant.

PDEs therefore are often more difficult to solve, and require numerical methods such as finite differences, finite elements, finite volumes, spectral methods, etc. or they may only have solutions for specific boundary or initial conditions.