# 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]}

## References

[1] | Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) |

[2] | Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 1.2) |

[3] | Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) |

[4] | Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 1.2) |

[5] | Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) |

[6] | Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 2.5) |