Physics in a nutshell

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# 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] Hans J. Weber, George B. Arfken Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) [2] Lothar Papula Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 1.2) [3] Hans J. Weber, George B. Arfken Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) [4] Lothar Papula Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 1.2) [5] Hans J. Weber, George B. Arfken Essential Mathematical Methods for Physicists Elsevier Academic Press 2004 (ch. 8.1) [6] Lothar Papula Mathematik für Ingenieure und Naturwissenschaftler Vieweg+Teubner 2009 (ch. IV 2.5)

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