The acronym ODE stands for Ordinary Differential Equation. The key word Ordinary refers to the fact that all of the derivatives in the equation are with respect to a single independent variable, even if there are more than one independent variables.

For example, the f(x,y) depends on both independent variables x and y. An ODE for f(x,y) would only depend on derivatives involving one and only one of these two variables such as:

p(x,y)\frac{d^2f(x,y)}{dx^2} + q(x,y)\frac{df(x,y)}{dx} + r(x,y)f(x,y)= g(x,y)

What makes this differential equation Ordinary is the fact that, although there are y's all over the place, all of the derivatives are only with respect to x.

ODEs come up in physics all the time. For example, the force on a block with mass m connected by a spring with spring constant k is given by F = -kx where x is the displacement of the block from its equilibrium position. This can be related to the acceleration of the block using Newton’s Law F = -kx = ma. This is secretly an ODE because, by definition, a = \frac{d^2x}{dt^2} = x'' which implies m x'' + kx = 0 and we seek a solution for x(t) where t is the time variable.

ODEs are defined in contrast to PDEs which are Partial Differential Equations. In PDEs, the differential equation involves partial derivatives of more than one independent variable. For example, a PDE for f(x,y) could have the form:

p(x,y)\frac{\partial^2f(x,y)}{\partial x^2} + q(x,y)\frac{\partial f(x,y)}{ \partial x}\\ + u(x,y)\frac{\partial^2f(x,y)}{\partial y^2} + v(x,y)\frac{\partial f(x,y)}{ \partial y}\\ + w(x,y)\frac{\partial^2f(x,y)}{ \partial x \partial y} + r(x,y)f(x,y) \\= g(x,y)

Perhaps the most famous PDEs in Physics are Maxwell’s Equations.

Knowing how to solve ODEs is crucial because, often times, a PDE can be transformed into a set of coupled ODEs using a technique called Separation of Variables, which is one of the most common ways to solve for an electric potential using Maxwell’s Equations.