Classification of Differential Equations

A differential equation is an equation that contains an unknown function and at least one of its derivatives.

Order of the Differential Equation

The order of the differential equation depends on the highest appearing derivative. For instance, if the derivative of the highest appering order is a second derivative, then the corresponding differential equation is of second order.^[1][2]

Ordinary and Partial Differential Equations

If the unknown function depends only on a single independent variable, then the corresponding differential equation contains only ordinary derivatives and the differential equation is therefore called ordinary. An example would be: \begin{align} \frac{d}{dt} f(t) + 2 f(t) = 0 \end{align} If there are however several independent variables (e.g. different time and space coordinates) and as a consequence partial derivatives instead of ordinary ones, then the corresponding differential equation is called partial. An example for a partial differential equation is: \begin{align} \frac{\partial f(x,y)}{\partial x} + \frac{\partial f(x,y)}{\partial y} = 0 \end{align} Both ordinary and partial differential equations appear in many branches of physics. ^[3][4]

Linear and Non-Linear Differential Equations

If the differential equation can be written in the form $\mathcal{L}(f) = g$ where $\mathcal{L}$ is a linear operator and $g$ is a function that is independent of the unknown function $f$, then the equation is called linear.

What was a linear operator again? Assume that there are two solutions $f_1$ and $f_2$ of the differential equation. An operator is linear if each linear combination of the solutions is a solution as well, i.e.: \begin{align} a \mathcal{L}(f_1) + b\mathcal{L}(f_2) = \mathcal{L}(a f_1 + b f_2) \end{align}

The ordinary derivative $\frac{d}{dt}$ is for instance a linear operator because: \begin{align} \frac{d}{dt} \left[ a f_1(t) + b f_2(t) \right] = a \frac{d f_1(t)}{dt} + b \frac{d f_2(t)}{dt} \end{align} Hence, an equation that is linear in the unknown function and all its derivatives is a linear differential equation.^[5][6]