I am just starting to read up on differential equations. The problem is (in my materials) nowhere is explained what do these dots mean. Can anyone shed some light?

$$\begin{align}&\begin{cases} \dot x=2x+y ,\\ \dot y=3x+4y. \\ \end{cases} \\ &\begin{cases} \dot x+x-8y=0, \\ \dot y-x-y=0. \\ \end{cases}\end{align}$$


1 Answer 1


It usually means derivative with respect to time. The notation goes back to Newton.

By the way, $\ddot{x}$ is often used in Physics for the second derivative of $x$ with respect to time.

  • $\begingroup$ so this would be the same as x'(t)? $\endgroup$ Jun 11, 2014 at 17:29
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    $\begingroup$ Yes, if by $x'(t)$ you mean the derivative with respect to time. The dot is kind of "reserved" for derivative with respect to time. In Physics, there will be plenty of derivatives not with respect to time. $\endgroup$ Jun 11, 2014 at 17:35
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    $\begingroup$ For example, the heat equation can be written $\ddot u=v^2u''$, which is the same as $\frac{\partial^2 u}{\partial t^2}=v^2\frac{\partial^2 u}{\partial x^2}$. $\endgroup$
    – Teepeemm
    Jun 11, 2014 at 19:35

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