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 $w$).
Answer
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.
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