If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable it is said to be an ordinary differential equation (ODE).

If different functions are differentiated with respect to different independent variables, but each function is only differentiated with respect to only one independent variable, is the equation ordinary?

That was a mouthful. For example, for this equation: $$\frac {dy}{dx} + \frac{dz}{dw} - 12y = 0$$

There are multiple independent variables ($x$ and $w$) but each dependent variable is only differentiated with respect to one variable (as opposed to say $y$ being differentiated with respect to $x$ and $w$). So would this be considered ordinary?

I know equations like this:

$$\frac {dy}{dx} + \frac{dz}{dw} - \frac{dy}{dw} +12y = 0$$

aren't ordinary because the one independent variable ($y$ in this case) is being differentiated with respect to multiple dependent variables ($x$ and $w$).

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    $\begingroup$ still a PDE. It is the presence of several independent variables that makes it a PDE. At least, this is my opinion. $\endgroup$ – James S. Cook Sep 6 '14 at 22:08
  • $\begingroup$ @JamesS.Cook Thank you. Is there a general consensus within the math community that the presence of several independent variables makes an equation an PDE or does it vary on a person to person basis? $\endgroup$ – dfg Sep 6 '14 at 23:00
  • $\begingroup$ As far as I have ever read, ODE means just one independent variable whereas PDE means several. Although, sometimes one considers families of ODEs which perhaps blurs the line a bit. The family is indexed by some other parameter which doesn't have the same role as the independent variable in the ODE... in any event, so far as I've seen yes my initial comment is standard. $\endgroup$ – James S. Cook Sep 7 '14 at 0:12
  • $\begingroup$ @JamesS.Cook Thanks. If you post your comment as an answer, I'd be glad to accept it. $\endgroup$ – dfg Sep 7 '14 at 0:43
  • $\begingroup$ Thanks! I felt guilty to post it without at least adding a link or two, so, there you go. $\endgroup$ – James S. Cook Sep 7 '14 at 1:40

A partial differential equation is one for which there are several independent variables. In contrast, an ordinary differential equation has just one independent variable. For example, see this discussion of PDEs or see this discussion of ODEs.

Often, in the standard examples of PDEs we separate solutions of several variables into products of functions of one variable. The one-variable functions are solutions to an ODE which is derived from the given PDE. This technique is known as separation of variables.

  • $\begingroup$ Thanks, appreciate it James! $\endgroup$ – dfg Sep 7 '14 at 3:37

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